Jerry is messed-up. But we can't really blame him—he has no apparent job, his parents died when he was a kid, and he lives in a dump. He keeps two empty picture frames in his room because "I don't have pictures of anyone to put in them," which is pretty much the most emo thing we've heard since Simple Plan was sorry they couldn't be perfect (113-115).
Also, he throws himself on a knife at the end of the play and kills himself. He is, in short, the sort of person who, if you saw him in the park, you'd want to go in the other direction at a rapid clip. When he saunters onstage at the beginning, muttering about zoos and such, everyone in the theater should shout, "Get out of there, Peter!" (It would annoy the actors, but sometimes that's what it takes to save a life.)
So why doesn't Peter hightail it out of there when Jerry shows up, anyway? Part of it is that you maybe can't tell instantly when someone is bad news (though random strangers shouting at you about the zoo seems like it should be a tip-off).
But part of it, too, is that messed-up is interesting. Jerry's unpredictable and dangerous, but that's why he holds our attention. "I find it hard to believe that people such as that really are" (145), Peter says, and though he's referring to Jerry's landlady, he could just as easily mean Jerry himself. Jerry doesn't fit; he doesn't work—and that's annoying but delightful, too, like being tickled until you almost pee your pants.
Both the irritation and the delight are uncomfortable, though. If Jerry is messed-up, it's because he doesn't fit where he's supposed to, doesn't conform to the "old pigeonhole bit," as he says (108). And a big part of why he doesn't fit is because people like Peter won't let him fit. He's poor, so he has to live in squalor; he's "h-o-m-o-s-e-x-u-a-l" (124), which he has to spell out, as if he doesn't want to admit, to Peter or to himself, what he's really saying. Peter has "everything in the world you want" (247), and Jerry doesn't have much of anything. If Jerry's an object of pity or amusement, it's because Peter has the power to pity him and be amused.
Throwing himself on the knife seems like it's Jerry's way of showing Peter that he's always been the one in control—the one who gets to decide what's normal and what isn't; the one with the bench. Jerry's the weird one, but you can't have "weird" without something "normal" to compare it to. It's like Jerry is Peter's dream—all the bad, scary, exciting things that aren't in Peter's life end up in Jerry's. And when Jerry gets sick of that, he gets Peter to kill him. Yeah, bet you didn't see that one coming.