Short Answer: The success- both immediate and extended- of The Simpsons made it much, much easier for other animated shows aimed at something other than children- although they universally had to be adult comedies- green lit by the suits.
Longer answer:
People tend to forget that cartoons are incredibly expensive to make- like, we've spent a century finding new and creative ways to drive cost out of them, some horrifying like Clutch Cargo which super imposed live action mouths over cartoon stills to avoid having to animate the human jaw, some just outright cheap like Hanna-Barbera having virtually all their characters wearing neckties so that they could more easily recycle cells to cut down on costs. Cartoons could certainly dial the quality up to 11- the original Superman Cartoon was one of the, if not the first piece of animation that made use of rotoscoping technology and the quality shows, but Superman also cost millions to produce, in Depression era US dollars.
As a consequence, you usually only see anyone with the money to finance it green lighting sure bet productions. That's why, for the longest time, the things that got produced were things that were already known to be valuable properties, and they were overwhelmingly targeted at children- you got spiderman and batman and superman because that got butts in seats and sold cheap plastic toys. Even riding on the cultural success of Lord of the Rings, Disney was unable to keep The Black Cauldron at their intended PG-13 rating (although that's a story unto itself). The Simpsons changed everything. Prior to that unless you were named Ralph Bakshi or maybe you were just going to slip something in because you were named Don Bluth you basically didn't see anything targeted at an adult demographic in the US, and unlike, say, a sitcom, there was no such thing as a shoe string budget. Animated productions you were in the tank for, for the better part of a season.
But then the Simpsons happens. It seems quaint now, but in an environment where your atypical family sitcom was a live action gig making a sickly sweet vertical slice of middle American life where you might have a single episode that deals with racism or alcoholism with kiddie gloves you instead had an image of middle America that upset then president Bush Sr. Late 90's meant that the Simpsons had spent enough time in continuous production that the mania surrounding it had died down, meaning that any new adult oriented comedy wasn't inherently going to get the 'Simpsons rip off' label, but had also meant that the Simpsons had spent enough time doing victory laps that TV executives eyes did that thing where they turn into dollar bills. You'd already seen some shows with moderate success- Duckman managed to stay on the air for four seasons- and that was on cable!- and The Critic was a successful, if critically mishandled property.
So by the late 90's you had an aggregation of talent with experience on other animated series, combined with a willingness to fund these projects. And because national TV networks were willing to fund them, you then had cable network channels like MTV and Comedy Network asking, "What about us?" Which is where you get South Park, Daria and Beevis and Butthead.
