The answer to this question will be different for every reader. In fact, some readers might resent being asked the question in this manner--it assumes that everyone does like something about the story, when in fact it's possible that a reader dislikes the whole thing!
What I personally like most about "A Summer's Reading" is that the protagonist is so flawed and human. He's awkward, insecure, and perfectly believable as a person. When I read about how George is so upset and ashamed that he's hiding in his room and the act of breathing feels like "drawing a flame into his lungs," and when I see how relieved he feels that "Mr. Cattanzara hadn't told on him," I really identify with him. Too many characters in short stories are so honorable, talented, and conscientious about everything that I get tired of them, and George is a welcome change from all that. His shame over his own behavior, and his relief at getting away with something, are easy to identify with. And when he finally heads to the library and selects a hundred books to try, I feel genuinely proud of him for trying to make a change.
Something that other readers might identify as their favorite aspect of the story is the almost magical way in which George's sense of self-worth is raised up from nothing, just because someone (Mr. Cattanzara) believes that George will accomplish a goal (reading the hundred books). When we read this part, we're reminded of how true that is: that even the act of setting a goal and sharing it with someone else can lift us out of our lethargy and give us the confidence to pursue the goal. For example, imagine you know you need to lose some weight; you post on Facebook that you're going to start running three times a week so you can lose ten pounds, and suddenly your friends are posting encouraging comments in response. You feel like you're halfway to your goal already because your friends believe in you! This same experience happens to George in the story, and it feels encouraging and authentic to read about it.
Still other readers might like the title of the story best, finding it funny and a little ironic. That is, the story is called "A Summer's Reading," but George spends the whole summer not reading. It's funny in the same way that the title of "Contents of the Dead Man's Pocket" is funny--because (spoiler alert) the main character doesn't even die in the story.
Finally, if loyalty and friendship are especially important values to you, you might particularly like Mr. Cattanzara and how he spread the false but kind rumors around the neighborhood about George having done all that reading. Maybe you've had a friend do the same thing for you, sharing possibly false information about you to help your reputation, or to make an embarrassing rumor about you disappear. If so, then your favorite thing about the story may be Mr. Cattanzara's kind actions and the way they buoyed George's sense of self-worth.