"The Story of Jerry and the Dog" is about Jerry's desperate attempt to connect with his landlady's dog, which tries to attack him every time he enters his building. The story is about the lengths to which Jerry will go to form some relationship with the dog, either as friend or enemy. He goes from feeding the dog to trying to poison it, but in either case, but the dog remains indifferent. The suggestion is that the world is immutable and unknowable, at least to Jerry, who tries so hard to leave a mark, but who always, in the end, is alone. The best he can do with the dog is achieve an uneasy truce: the dog simply ignores Jerry. As Jerry says at the end of his speech: "I have learned that neither kindness nor cruelty by themselves, independent of each other, creates any effect beyond themselves; and I have learned that the two combined, together, at the same time, are the teaching emotion. And what is gained is loss."
In another sense, "The Story of Jerry and the Dog" is simply a recapitulation of the larger story of Jerry and Peter. Jerry is trying to make some impression on Peter, like he did with the dog; the games Jerry played with the dog are similar to the ones he plays with Peter in trying to draw him out. In this case, the meaning of the story with the dog becomes larger than a simple anecdote -- it becomes a kind of parable for modern life.