Part of the story of the 'wacky morning DJ' is in fact the story of the DJ itself. Generally, when we think of radio, we think of either two basic formats: a DJ playing recorded music (DJ standing for 'disc jockey', i.e., someone who played discs on a turntable to make the music happen) or talkback. But it's worthwhile to remember that, in a time before the mass uptake of TV in the 1950s, that a lot of what is now on television - drama, sitcoms, detailed news broadcasts - was originally a function of radio. It's also worthwhile remembering that the idea of playing a recording on radio was still quite foreign as late as the 1930s - before that, music on the radio was mostly played live (and indeed, in the UK, live performances were a staple of the BBC for a very long time - in 1963-1964, the Beatles seemed like they performed on some show recorded at BBC studios basically every week, and they were merely one in a chain of different performers who would do this). The idea of a DJ playing recordings had to wait until the mid-1930s, as equipment improved and the limitations of recordings gradually decreased.

In 1934, a radio announcer at the Warner Brothers-affiliated KFWB in Los Angeles, Al Jarvis, had the idea for a 'Make Believe Ballroom', where, using recordings, Jarvis 'made believe' that he and his listeners were in a ballroom dancing together to the (swing music) pop music of the day. Jarvis' show became rapidly very popular - to the extent that Nat King Cole had a song called 'Jivin' With Jarvis' - and was widely imitated across America.

According to a paper by R. Serge Denisoff, one of the strengths of Jarvis' format was that he was an innovator in terms of not simply being a 'radio announcer' but instead a 'personality'.

Radio announcers, prior to the arrival of Al Jarvis to KFWB, Los Angeles, were aptly described as "The same staff spieler who read poetry announced each disc solemnly, impersonally, and formally enough to qualify as an adept funeral director." Jarvis personalized the announcer's role with the addition of a conversational and friendly microphone style. The approach, perfected by Arthur Godfrey in the East Coast, elevated the announcer from a mere stylized reciter of poetry ads, and record introductions into a "personality" who commanded the attention of his listeners, thus adapting as equal* if not superior, status to the music he played.

Of course, I suspect if we were to listen to Al Jarvis today, he wouldn't exactly come across as, say, Crazy Ira and the Douche (the 'wacky morning DJ' parody on Parks And Recreation) but the idea of 'wacky' is of course subjective and bound to the conventions of an era; to be wacky is to flout those conventions in some kind of way, and those conventions change over time.

In the US in the 1950s (and elsewhere), radio formats changed dramatically as TV took over many of the previous functions of radio. Why listen to a drama when you could watch it as well? The Buggles, of course, chronicle this changeover in their song 'Video Killed The Radio Star' - "I heard him on the wireless back in '52". This led to a dramatic increase in the importance of the Al Jarvis-style DJ to the radio stations of the era. The things that radio basically still did better than TV were fundamentally audio - basically anonymous people calling into the radio station on their telephones, or just playing music. This expansion of the importance of the Al Jarvis-style DJ led to DJs trying anything they could to grab a hold of a market and exploit it (for not only acclaim but profit - this was also the era of the most blatant payola, the practice of record companies providing lavish treats etc to DJs to play their records). And one of the things those DJs did during this era was to start to promote 'wacky' personae, using their voices and words to portray certain vibes. Alan Freed with his Moondog shows was an early innovator in this regard, attracting a very strong, devoted teen following in Cleveland and then in bigger markets, to the extent of playing a strong role in the rise of 'rock'n'roll' as a genre aimed at white teens.

The rise of rock'n'roll in the 1960s - the Beach Boys, The Beatles, etc - was further associated with DJs with increasingly out there radio personae, usually broadcasting under nicknames of some description. Most notable are 'Murray The K' (a 1964 recording of which is here), 'Wolfman Jack' (you can hear a 1960s recording of his Howlin' Wolf influenced vocal personae in the here), and 'Cousin Brucie'. Generally, across the late 1960s and 1970s, there was a sort of war between the radio broadcasters and the DJs. Commercial radio broadcasters became increasingly convinced that the wilder more eclectic radio shows - usually the province of the big personality announcers - were bad business (perhaps not necessarily just because of ratings, but because they were harder to control). Such executives strongly pushed for much more control over what DJs played, and what they announced and how. As a result, people like Wolfman Jack, in the 1970s, seemed like refugees from commercial radio in some way, either broadcasting on the more experimental FM stations, or appearing as introducers on TV (in Wolfman Jack's case, on The Midnight Special).

Obviously, however, what prompted this question is that a breakfast radio format that is focused as much on comedic hi-jinks as much as music, and which often features comedians riffing off each other. As OP says, it's pretty ubiquitous - you likely don't have to scan the airwaves for long to find a local version. As radio station playlists got increasingly automated and the job of the DJ became increasingly about being a voice on the air, it's perhaps unsurprising that DJs looked to expand that role. This comedy + music radio format developed in the early 1980s is usually called 'Morning Zoo'; an October 1984 Los Angeles Times article about the format said that:

The original "Morning Zoo" was created by program director Scott Shannon at WHTZ-FM (Z-100) in New York City more than a year ago, and through a wacky mix of music, conversation and comedy that brings back memories of the zany Top 40 men of the 1960s, the Top 40 station has made it all the way to No. 1 in the nation's largest radio market. Since then, morning zoos have been instituted at Top 40 stations from Los Angeles to Boston, using the format's traditional reliance on strong, dominant personalities who give their listeners not just music, but entertainment.

Broadly speaking, if variations on 'Morning Zoo' (so named because WHTZ at the time was often called 'The Zoo') has dominated radio formats since the early 1980s - with different breakfast radio 'teams' going for different kinds of discussions and comedies within the basic format - it's for the obvious reason: it continues to rate well. Why this was the case in the 1980s but not the 1970s to the same extent might be to do with cultural shifts more generally. But part of the story is likely that, especially from the 1980s onwards, especially, radio started receiving competition from, well, listeners with their own playlists. Cassette players and then CD players became ubiquitous in cars (remembering, of course, that plenty of people listening to morning radio are driving to work - in contrast, you'd have to be brave to play a vinyl record in a car, what with all the bumps in the road), and these formats of course had longer play times and needed less adjusting than a succession of 7" vinyl singles or a even 12" record (that you have to change every 20 minutes rather than every 45 minutes for a 90 minute cassette, which even automatically changed sides in many systems). Those who wanted to listen to their own specific musical tastes could make cassette mixtapes or burn CDs of their own (and then of course, in the 21st century, you could put your iPod on 'shuffle' or go to a Spotify streaming playlist). This meant that the radio station just playing music had to adapt, to provide something that a mixtape could not provide - which was of course, comedians being funny, live on air.

References

Laurence W. Etling (1999) Al Jarvis: Pioneer disc jockey, Popular Music & Society, 23:3, 41-52, DOI: 10.1080/03007769908591744

B. Lee Cooper (1990) From Anonymous Announcer to Radio Personality, From Pied Piper to Payola: The American Disc Jockey, 1950–1970, Popular Music & Society, 14:4, 89-95, DOI: 10.1080/03007769008591416

R. Serge Denisoff (1973) The evolution of pop music broadcasting: 1920–1972, Popular Music & Society, 2:3, 202-226, DOI: 10.1080/03007767308591013

Talbot, Mary (2012) "Will you sing along, Tina?" Zoo Format and Women's Place on Radio One. Culture, Society & Masculinities, 4:2, 163-174.