Jerry is a loner, and he is trying to get Peter to understand loneliness and suffering. Not necessarily Jerry's own suffering, but the suffering of the animals in the zoo, the other tenants in his building, about the rest of the lower-class members of society of which Peter ignores, or tries not to think about.
Jerry tries to tell Peter stories that he thinks will disturb him, in order to teach him about his life, and the lives of others like him. It doesn't quite work, because although Peter is interested, but he isn't moved, he is more repelled and bewildered.
In Act 2 of Peter's life, Jerry wants him to consider these people, to engage in communication with them, to not put them out of his mind, as he did before. Jerry has changed Peter's life forever,