It's because banana peels came to symbolize the growing problem of trash-filled streets in late-19th-century American cities, and they became the go-to reference point Vaudevillian comedians used to satirize the issue.
From here, the origin of the reference splits in slightly different directions, but they're all still centered around people in major cities slipping in their own filth. In one proposed origin, a banana peel is symbolic for a much less stage-friendly real-life problem: mounds of s**t in the streets. Literal s**t. Between the trotting horses and the wild pigs that once roamed free, the streets of New York were a giant septic tank. Comedians wanting to reference it in their acts were a bunch of pussies who didn't believe in the true meaning of art, so they substituted a banana peel for poop, because those, too, were everywhere.
That leads me to the second possible origin: Everyone tossed their trash on the streets because, to hell with everything; let's all just live in a rotting wasteland of our own garbage. Bananas were gaining in popularity and people were disgusting, fancy-dressed animals, so they'd toss their peels on the streets along with everything else.
Either way, the lovable doofus on stage who slipped on a banana peel wasn't just slipping on a banana peel. He was slipping on biting social commentary about the sorry state of 19th-century waste disposal.