What's Jerry up to and why?
The "why" of this question is more complex than the "what." It is part of what makes "The Zoo Story" so compelling.
Jerry never reveals a true motive. Is it that he is envious of Peter's lifestyle? Peter reveals to Jerry that he lives in the affluent East Side and Jerry tells him he resides in the slum-ridden West Side. Peter has a family; Jerry does not.
Does he want fame? Jerry makes a strange comment about the evening's headlines that is never truly answered. Perhaps he wants notoriety and is anticipating the reaction and attention committing murder will bring to him, even if from afar.
Is he simply psychotic or a severe personality disorder? There is evidence to support this argument as well. There is no reason for the psychological cat and mouse game, but plenty of dialogue to show Jerry enjoys the pursuit of his prey.
Jerry, the antagonist in The Zoo Story, confronts Peter while he is reading a book in Central Park and coerces him into partaking in an act of violence. Throughout the course of the play, Jerry tells Peter only what he wants Peter to know, and does not like to be asked questions or be judged. Jerry uses all of his resources, including his storytelling ability, his humor, and finally his violent aggression, to make-sure that Peter does not leave until he gets what he wants from him. Jerry finally gets Pete to engage in a fight with him, but ony because Peter had to do it in self-defense.