In America, most of this has it's origins within traveling carnivals. Catch-as-catch-can came to the US sometime in the late 19th century. They had 'shows' where they would have guys that were usually already trained in wrestling, or have athletic background. They would give the wrestlers various personas to attract people to the shows. The shows were 'fixed' but no one knew that at the time and people gambled on them a lot.

There were different classes of wrestlers used for different things. Sometimes during a show, they would issue a challenge for someone from the crowd to try and pin one of the wrestlers. In this case, 'hooks' were used (illegal in amateur wrestling) and could end the bout in seconds. 'Hookers' were used to protect their end of the business.

This is actually a really brief answer and there's a lot more to it, but this is how it started to evolve into the business of professional wrestling as we see it today.

This question can either refer to in-ring style, or promotional style. I will endeavor to take a shot at bout.

The theatrical style, from a story line perspective, always existed, going back to the earliest days of what we deem professional wrestling. The theatrics associated with both bare knuckle boxing and wrestling as challenge matches (hat in the ring, sniping at each other through articles in a local paper, etc.) were common place in both sports. Boxers and wrestlers both would hold exhibition matches to promote the "real" matches, sparring common opponents over and over. These exhibitions would lend themselves to hype, with actual boxing and wrestling matches ending in draws or no contests, with follow on matches (and associated exhibitions) to earn more money.

Frank Gotch, who was the first American to hold the Heavyweight Championship and and, whom, according to Nat Fleisher, "...was to wrestling history in this country what John L. Sullivan was to boxing. He dominated the field." In Mark Hewitt's article Wrestling in the Klondike, Frank Gotch was using a classic pro wrestling trope in 1901, as seen below:

Before and after the Yukon trip, both Gotch and his manager Joe Carroll, who was then using the name Ole Marsh, were members of Farmer Burns' troupe. While there are some indications that Gotch was simply a happy-go-lucky farm kid out on a lark, Carroll, perhaps with Burns' arrangement, was pulling what was known as the "badger game." In this oft-used pro wrestling scam of that era, Carroll would show up in a gold rush boomtown such as Dawson City in the Yukon Territory and proclaim himself "the champion of the Yukon." He would then orchestrate a series of challenge and "grudge" matches between himself and other wrestlers, in this case Col. J.H. McLaughlin and Gotch. McLaughlin had been a pro wrestler since the post- Civil War period and claimed the world collar and elbow championship since 1870. Gotch meanwhile was the "ringer." His cover was that of "Frank Kennedy" of Springfield, Missouri, and he was a Filipino-American War veteran prospecting for gold.

Carroll actually went to jail in 1910 for his role in match fixing in both boxing and pro wrestling.

As for the theatrical ring style, that was Gus Sonnenberg. Sonnenberg as one of the first crossover stars in wrestling, coming from the Providence Steam Roller (1928 NFL Champions). He tried his hand at wrestling, at a time when most wrestlers had only trained for their sport, finding success due to the flying tackle. From Steve Yohe's masterful Ed Lewis book:

His style was to work everything around a move called The Flying Tackle. He would fly across wrestling rings, like he did on football fields, to drive his head into the chest and stomachs of his opponents. He was quick on his feet and could deliver a series of tackles before the other wrestler could recover from the first. The true finisher was a tackle where he grabbed his opponents two legs and combined the football move with a double leg take down. The other wrestler would crash to the mat hitting the back of his head on the canvas. When he missed the move, he would fly out of the ring into the crowd or to bump on to the arena floor. In a way, Sonnenberg was the first wrestler in 3D. A fan or news reporter at ringside had to keep their head up because at anytime Gus could end up in his lap.

Sonnenberg revolutionized pro wrestling. From the beginning of time, the major stars were defensive wrestlers like Gotch, Londos and even Stecher. With Sonnenberg, his offense was his defense. He was aggressive and never let up. There was action from the first minute to the last, and the fans loved the style. Gus's matches were short and sweet. His style was a result of the fact that he wasn't a 'wrestler'. He was an athlete and a 'worker'. Any mid-level true wrestler could have beaten him in a contest. But Gus was very tough and physical and the style could be called 'stiff' but the idea was to put on a good show and entertain.

I could (with some time and specificity) be able to find exactly when certain theatrics were introduced to bare knuckle and early wrestling, but the general concept is as above.

That really depends on what kind of American pro wrestling you're talking about

If you're talking about stuff like Hulk Hogan with his overly theatrical boot and legdrop i personally would say it somewhat started already in the 1950's when TV slowly but surely started to pick up on Pro Wrestling as a show rather than a sport.

One of the first real theatre style gimmicks would have to Gorgeous George with his very narcissist character.

The "Golden Age" of wrestling as it called didnt last very long though, as wrestling slowly started to fade away, people didnt seem to like the (still) gritty matches and wrestlers that didnt look like a sportsman.

Then along came Vince McMahon with (then) WWF (or WWWF, depending on how far you go back).

Vince had a plan, he had bought out his dad for the New York promotion and he wanted to spread it from just the New York area to the whole world.
He got wrestlers training and wrestling slowly started picking up again.

By the 1980's wrestling was a smash hit, Vince then coined the term "Sports entertainment".

Hulk Hogan came to WWF in 1979 with a new gimmick of All-American, with bright colours really hit with the people, kids adored him, men wanted to be him and women wanted to be with him.

With stuff like "Rock 'n' Wrestling connection" where they mixed wrestling with musical elements only helped with putting wrestling on the map as a serious competetitor to other entertainment outlets like TV shows and movies.

In 1985 the world saw the first ever Wrestlemania (On closed circuit) with Hulk Hogan and Mr. T as the main draw.
Entertainment fans wanted to see Mr. T
Wrestling fans wanted to see Hulk Hogan.

Also in 1985 Hulk Hogan was featured in "Hulk Hogan's Rock 'n' Wrestling which is an animated TV series, so even more kids watched.

Pro Wrestling went steady, a bit lower ratings, over the years untill the mid 90's with the Monday Night Wars between WWF and WCW.

Many of the 80's stars of WWF had jumped ship over to WCW.

WCW with their Monday Nitro and WWF with their Monday Night Raw competed each monday for TV ratings.
WCW pretty much led the TV battle between 1995 and 1997.

In 1998 we saw Vince McMahon as a on-screen talen (Known as Mr. McMahon) feud with Stone Cold Steve Austin.

Stone Cold has always been portrayed as a working class man, so people would cheer for him, especially while he beat his boss, Mr. McMahon.

With various celebrity guests on both shows (Including Mike Tyson) wrestling was a must watch programs for many young people.

Pro Wrestling had gone from kid friendly Hulk Hogan of the 80's to DX with their famous crotch chop and "suck it".
Of course not everyone liked the direction it was going but most people seemed to.
Who wouldent want to stick it to their boss??

After the Monday Night Wars ended (With WWF buying out WCW) they needed a new roster so the roster split happened.
Around half of the roster stayed on Raw while the other half went to SmackDown.

In 2002 a lawsuit filed by WWF (the panda one) made it so that they couldent use that name, so they changed it to WWE.

From around 2002 to 2007 WWE entered the "Ruthless Aggression" era, which brought up a lot of stars like John Cena and Randy Orton.

WWE really cleared up the program, it became more kid friendly.
WWE became rated PG.

There is (and where) a fair bit of comic characters, like Brodus Clay, the dancing guy from Planet Funk.

The modern pro-wrestling style you see, with Irish whips off the ropes, powerslams, suplexes, arm drags, chops; was invented by the Gold Dust Trio in the 1920s as a way to add more excitement to the slower, grappling style that Frank Gotch popularized.

From WWE Legends

It started in part from Mondt's desire to remake the sport into something it had never been before. Up to that point, matches normally went on for hours and consisted of very little movement or action, with opponents exchanging in subtle wrestling techniques that often bored crowds. Furthermore, matches were made in haphazard fashion, with very little buildup. After convincing Lewis and Sandow, Toots would change all this.

He introduced what he called "Slam Bang Western Wrestling," a style that incorporated striking, kicking, and other types of blows-in other words, maneuvers that were not really "holds" in the true sense. This created more action and excitement. He instituted time limits, which caused wrestlers to speed up their matches. Entertaining the crowd became top priority. In order to properly hype the matches presented by the Gold Dust Trio, Mondt created the concept of the "program," an ongoing feud that would generate interest and help build up new stars. For the first time, competitors were divided into clearly defined "good guys" and "bad guys." This was made even easier by the Trio's idea of presenting wrestling for the first time as a "packaged" touring attraction, with an established troupe of competitors appearing together in each city. The sport has remained more or less in the same form ever since.