Jerry keeps telling Peter that he's going to go into detail about what happened to him (Jerry) at the zoo before he came to the park. He never manages to get it out, though. So The Zoo Story is a story we never hear. The title is a kind of joke or tease; another way in which the play refuses to quite make sense; it's about frustrating you, rather than making you happy with action or adventure or a love story or even a pleasant trip to the zoo.
The Zoo Story also refers to the play itself, though; it's a story about a zoo. Jerry and Peter are animals on display for your enjoyment ("You're an animal too," Jerry tells Peter at the end of the play (278)). And as with animals in the zoo, it's not always easy to tell why they're snuffling over there, or what they're thinking. The story at the zoo is a story you can't quite understand—not the least because, like you, the animals you're watching are animals, too.