Raymond is obsessed with the "Who's on First" bit from Abbott and Costello . It's kind of like his security blanket; he recites it when he's feeling nervous or tweaky. The thing is, he doesn't understand it's a joke—that the question "Who's on first?" isn't really a question at all (since there's a dude named "Hu/Who" who is, in fact, on first).
Charlie remarks that if Raymond could just understand the joke, he might "get better." Of course, that's not really the way autism works, but he's right that if Raymond were able to get the nuance of that joke, that would mean his awareness of social cues/humor had gotten a lot better…
So, in short, that bit really becomes a big fat symbol of the struggles Raymond has in operating in the world, particularly in terms of understanding people and interacting socially, and of his/Charlie's efforts to overcome them.
At the end of the film, when Ray is about to get on the train to go back to Wallbrook, you can tell that he's come a long way when he's able to tell Charlie that the skit is "Very funny."
You don't get any real sense of whether he truly "gets" the humor there, but at least he knows it's supposed to be funny—and that's clearly a huge step from using that endless series of questions—questions with no answer!— as a security blanket.