So the first "punk specific" dancing was "the pogo", just jumping up and down (though you had Iggy Pop doing an early form of "crowd surfing" as part of the Stooges in the early 70's). There is no definitive story of where pogoing originates, but I've seen people credit it to both sides of the Atlantic, with some even crediting it to Sid Vicious before he was in the Sex Pistols.

Moshing has been credited to multiple scenes, but all part of the broader American hardcore movement. Chronologically, it seems to make the most sense that it developed in California along with the first generation of LA Hardcore (the one based in the city, with the Germs, X, Fear, Circle Jerks, etc. before the scene became slightly more suburban with Black Flag, Bad Religion, etc.). I believe you can see it in the The Decline of Western Civilization, which documents the LA scene and was filmed 79-80. It seems to have spread nationally very, very quickly and already in 80-81 you see kids in the East Coast scenes (Boston, DC, New York) doing it by all accounts.

The name also has an unclear history, but the most convincing explanation I've heard is that it was taken from Bad Brains singer H.R.'s pseudo-Jamaican pronunciation of "mash" (I believe this is in Blush's American Hardcore).

It seems to have passed into the alternative metal scene in the mid-80's, likely through both the "Crossover" bands (generally hardcore punk musicians who had become slightly more competent at their instruments, especially from the East Coast scenes) and the very hardcore-influenced Thrash Metal. By the late 80's, there was some backlash against in some circles of the hardcore movement--Fugazi, for example, famously would stop shows if people were dancing too violently. It was often framed as a specific manifestation towards the "tough guy" subcurrent within the movement.

Now, I'm not one to general put psycho-social explanations on things such as anxieties about the Cold War and the Reagan/Thatcher Eras, but I do see it much more as the natural extension of the development of punk rock. The very heterogenous arty New York Scene based around CBGB's and Max's Kansas City (the Ramones, New York Dolls, Blondie, Television, the Talking Heads, the Dictators, Patti Smith, the Velvet Underground, etc.) gave rise to the more homogenous but still diverse, much more working class British Punk scene, often called "77 Punk" (Sex Pistols, the Clash, the Damned, Sham 69, the Undertones, but also stranger bands like Wire, the Fall, X-Ray Specs, Crass, Gang of Four, and the Jam) who were influenced by the more traditional rock-oriented of the New York Punk bands (especially the Ramones, the New York Dolls, the Dictators, and Television) as well as pub rock and indigenous British rock bands like the Who and the Small Faces.

When these records start getting to America, it inspired a whole new generation of bands. Just as the UK sounds stripped down the New York sound, the first generation of hardcore from LA stripped down the UK sound. Though there were still arty bands like X in the first generation, younger hardcore bands (coalescing into the "Hardcore 81") took mainly after the Ramones, Sex Pistols, Clash, the Germs, Bad Religion, Black Flag, and the Dead Kennedys (from the Bay Area). Though there were still weird arty hardcore bands in the more suburban second generation of Southern California Hardcore (the Minutemen, for example), this didn't end up becoming part of the "core package" of hardcore that get spread out quickly to other cities which emphasized simply faster and louder and often but not always simpler and messier. You get DOA from Vancouver, Minor Threat and Bad Brains, Minor Threat many others in DC, the whole NYHC movement, etc. This stripping down is not only musical but aesthetically. You see fewer mohawks, etc. (especially on the East Coast where "Hardcore" continued more as a separate movement from "punk"), you see more sweatshirts and jeans and t-shirts. The stripping down of movement at the shows to the most elemental ordered but disorganized mashing/slam dancing seems like a natural extension of the changes from CBGB's New York to 77 UK to Hardcore 81 (which was increasingly male and increasingly suburban). UK's "UK 82" and Streetpunk also in the UK coalesced around a different set of symbols (UK 82 huge mohawks, Streetpunk being "lads", and both carried strong working class politics), but shared with their American cousins in Hardcore 81 the same narrow, stripped down sense of what sorts of things the movements contained. Both were also more male dominated, louder, faster, and less engaged with urban art movements than the first generations of punk and even LA hardcore had been.

Though you'd find moshing and crowd surfing at punk, hardcore, and underground metal shows in the 80's, it only really went mainstream when Nirvana broke and ushered in the brief success of "alternative rock" after 1991 (again, we see some of the more art-y groups apprehensive about moshing--I remember seeing a clip of Billy Corgan of the Smashing Pumpkins on MTV News in the mid-90s where he announced he was "talking a stand against moshing").

The TL;DR though is it became part of the "hardcore package" that spread through a variety of touring bands, most notably Black Flag. From there it spread to other genres. It's really hard to overestimate the influence of the early LA on the development of American punk and hardcore. I can't find the review now, but there's a great review that points out the irony of the first generation of Boston punks calling their main compilation This is Boston, Not LA because the sound was so much ripped off from LA bands like the Germs. For several years, the main constituents of American popular alternative were the commercially successful New Wave groups, arty "college rock" (groups like REM and Sonic Youth), and hardcore bands, most of followed a pretty clear LA template 79-~84.Husker Du's sprawling Zen Arcade, the Minutemen's eclectic Double Nickles on the Dime, and Black Flag's very slow My War, all released in 1984 and all on SST, the most powerful hardcore label at the time, were big records that really shattered the existing hardcore template. Blush argues by '86, the hardcore scene was essentially dead. There were of course bands following different templates (Big Black, the Bad Brains), but those three (new wave, college rock, hardcore) are often seen as the main trends of American alternative music in the early 80's.

Bibliography:

For a few of the big bands (Minor Threat, Black Flag, the Minutemen, Husker Du), Our Band Could Be Your Life is the best resource. For early New York, Please Kill Me. For what I call "Hardcore 81", Steve Blush's American Hardcore. For the early LA scene, We Got the Neutron Bomb. For the DCene, Dance of Days and the liner notes for various Dischord releases like 20 Years of Dischord. Some of the interview's in Punk Planet's book We Owe You Nothing are full of great history, too. There are further books I haven't read, like We Got Power! and Kids of the Blackhole (both focusing on the Southern Californian punk and hardcore), NYHC (focusing on the pre-Youth Crew New York scene, it seems), Girls to the Front (about Riot Grrrl), England's Dreaming and Punk Rock: An Oral History (both about the British scene, especially the first generation), Gimme Something Better (Bay Area). For hardcore into the 1990's (as opposed to the punk revival of bands like Rancid or Green Day), Burning Fight is the best book I've heard of (Steve Blush unceremoniously ends around '86, and doesn't do a good job capturing even slightly later developments). I haven't read Burning Fight, but during undergrad, our school's "punk rock society" organized a serious of lectures (which we called "the Common 'Core") and we had the author come and give us a talk when the book was still in draft form. There are a few later social science books on punk, Haenfler's Straight Edge (which is partially an ethnography in Denver, if I remember) and Wood's Straight Edge Youth. No bibliography of punk rock would be complete without Hebige's book Subculture: the Meaning of Style, which discusses early UK punk and mod scenes.

Bonus: an older thread--What happened to punk culture?

Edit: I want to note, this is more of a Sociology question and specifics of the history are relatively unknown thus I am trying to give a contextual image to try and answer your question. I apologize if my tier 1 response is not up to /r/AskHistorians standards.

There isn't a specific date or instance that I know of but hardcore dancing, slam dancing or moshing came from the Harcore Punk scene of the early 1980's. The mode of dance fits in with the rebellious, angry, chaotic and antisocial/anti-establishment nature of the early scene (think Black Flag, Circle Jerks and Dead Kennedys) which sprouted from discontent and disillusionment of American exceptionalism that was a hallmark of the Reagan-era. The youth that participated in the hardcore punk scene were the generation that arose from the 60s-70s "Flower Child" protest era that was in someways seen as a failure for not creating "real" social change. This lack of seeing progressive social change and the Cold War seemingly becoming more volatile (Nuclear war became a real possibility at this time) with Reagan pushing hard against communism and leftism in general, turned the peaceful protest music of the 60s/70s into one one of hopeless anger and rebellion against not just the political structure but the social structure that allowed for the status quo to exist. The angst of the music and slam-dancing became a way of shedding their content with what they felt were unjust social situations that they felt they had no true control over and thus a social outlet for releasing their anger. Patrick J. Williams in "Subcultural Theory: Traditions and Concepts" states that slam-dancing was a way to parody the seeming growth of acceptance of violence in American society that was becoming more glorified by movies/television.

The dance form grew becoming a part of the growing Extreme Metal scenes as well as becoming a centerpiece of hardcore punk shows but lost it's contextual meaning as it became a tradition for shows of these music scenes and now just plays into the angst of their aesthetic.

In the book Our Band Could Be Your Life: Scenes From the American Indie Underground 1981-1991, Michael Azerrad writes:

When it started, hardcore dancing was only stylized violence, an expression of the clamor and aggression of the music, and of the participants' adolescent inner tumult. "It was a lot of fun," said Dischord [Records, a seminal D.C.-based hardcore/punk/indie label founded by Minor Threat's Ian MacKaye which still puts out excellent records to this day] stalwart Cynthia Connolly of a typical early Minor Threat show. "It was a group of people having a real good rowdy time."..."It was completely a reaction to the Bump, the Hustle, all these fuckin' dances," MacKaye explains.

To see how we got to that point, let's explore some background.

By 1981, punk was considered a dying genre. The Sex Pistols broke up in '78. The Clash had started incorporating new styles on London Calling and Sandinista! (1980) moved the band even further from their punk roots by incorporating dub, reggae, r&b, roots rock, and even rap. Even the Ramones had released their best albums already and had enlisted Phil Spector to produce End of the Century in 1980 in a bid to create a more polished, mainstream album. Many of the early fans and band members had moved on to arty post-punk like Television, Wire and Mission of Burma or new wave like Joy Division and Siouxsie and the Banshees. Some of the early bands evolved into a more mainstream sound such as Talking Heads and Blondie.

The inauguration of Ronald Reagan in 1981 also seemed to herald a new age of American conservatism. Hardcore actually sprung up in the well-to-do, Reagan-supporting suburbs surrounding Los Angeles and Washington D.C. as a reaction to the conservative milieu the kids were growing up in. Writer Barney Hoskyns said the new music was "younger, faster and angrier, full of the pent-up rage of dysfunctional Orange County adolescents who'd had enough of living in a bland Republican paradise." Hardcore was essentially the style of early Ramones, Clash and Sex Pistols distilled to its essence. Kids with limited music ability were playing punk as fast and loud as they could and infusing it with a raw anger that was markedly different from the political anger of early punk bands (although hardcore certainly had political elements to it). To give you an idea what kind of anger we're talking about, there's this quote from Henry Rollins:

There's a lot of people with a lot of fury in this country - America is seething at all times. It's like a Gaza Strip that's three thousand miles long.

Black Flag was the flag carrier for hardcore with roots in a band called Panic formed in Hermosa Beach, California in 1979. Panic was formed by Greg Ginn, Keith Morris and Chuck Dukowski who formed the core of Black Flag. The band found gigs at Hollywood venues tough to come by because the older bands and fans in the city disdained the amateur, working-class style (jeans, t-shirts, sneakers, long hair) and sound of the bands from the suburbs, favoring the safety pins, studs, mohawks, leather jackets and sounds of 1977-era British punk. Instead of traditional gigs, the band started playing parties around Orange County which inspired a fast style of short songs that allowed the band to play as many songs as possible in as short a time as possible. Greg Ginn said this about it:

That's where we really developed the idea of playing as many songs in as little time as possible because it was always almost like clockwork - you could play for twenty minutes before the police would show up. So we knew that we had a certain amount of time: don't make any noise until you start playing and then just go hard and long until they show up.

This new style of punk quickly attracted the attention of the cops who routinely broke up shows which degenerated into near riots. The kids tended to be bigger, more athletic and less drug-addled than older punks and showed a willingness to fight back against the police. At least a dozen Black Flag shows ended in violent confrontations between punks and police in 1980 and 1981. Soon, hardcore bands were banned by many LA clubs. An editorial in an LA paper was titled "The Black Flag Violence Must Stop!". Black Flag eventually started attracting people looking for violence, which the band did nothing to discourage. Henry Rollins said:

One of the main rallying war cries for us was "What the fuck, fuck shit up!" Literally, that was one of our slogans.

The band even went so far as to create radio ads for their shows in which they mocked the LAPD, including Police Chief Daryl Gates by name. Their shows were loud, fast, and violent, especially after Henry Rollins joined the band as lead singer. Rollins had attended a military academy while growing up in D.C. and was an angry, violent, physically intimidating person with a charismatic stage presence. He often got into fights or ended up bleeding at shows. Rollins wrote in his tour journal Get In the Van about a German show:

I bit a skinhead on the mouth and he started to bleed real bad. His blood was all over my face

In Vienna, Rollins was hit in the mouth with the mic, spit on, burned with a cigar, and punched in the jaw before cops showed up. The cops were beaten up, had their uniforms stolen, and allegedly had their police dog killed.

The other important band for the evolution of hardcore and moshing is Minor Threat. Minor Threat formed in Washington D.C. having been heavily influenced by Black Flag. Minor Threat shows, like Black Flag shows, were fast, angry and violent. Ian MacKaye said:

I was nineteen...and I was furious - the evolution of punk rock had grown from these silly kids to being embattled silly kids to being embittered silly kids, then embittered kids and then violent, embittered kids. It just kept getting ratcheted up.

The fans basically followed the violence of the band, music and police. The shows were often fight-filled anyway and there's not a huge jump from fighting to slam dancing. Moshing was yet another release of violent energy and tension that matched the energy of the music. Hardcore shows became cathartic bombs of energy being released and mosh pits were a natural way of "dancing". Furthermore, as Azerrad writes,

Hardcore punks were happy, even determined, to limit their appeal. The music was resolutely unmelodic, humbly recorded, and vastly unsexy. It was a point of honor not to reach out beyond their own nationwide tribe. It was not only a way to cement a fledgling community, but like slam dancing itself, it was also a way to feel powerful at a time in life when one can feel particularly powerless

Unfortunately, over time, hardcore's violent reputation, fed in part by slam dancing, began to attract people who were interested merely in the violence and turned the scene from a way for friends to release pent up aggression into a hostile, fight-filled environment. The varying scenes spawned gangs that often beat up bands that came in from outside the area, moshing crossed the line into straight up fighting and the scene started to disintegrate as earlier, core scene members began to drop out. Black Flag moved towards metal starting with My War, Minor Threat and other vital early bands like the Faith broke up and the first wave of hardcore began to peter out by 1983 or 1984. As Ian MacKaye said "I'm tired of being run into by fucking marines."

Sources: Our Band Could Be Your Life: Scenes From the American Indie Underground 1981-1991 by Michael Azerrad, Get In the Van by Henry Rollins

https://en.m.wikipedia.org/wiki/List_of_classical_music_with_an_unruly_audience_response