What causes Jess to smile at Leslie in Bridge to Terabithia?
The part of the story that the question is asking about can be found in chapter four. The start of the chapter focuses on the first day of school for Jess and Leslie. Jess is super excited to race at recess against the other boys, because he believes that he will be the fastest. Despite being a girl, Leslie shows up to race too. The boys are not happy about it, but they let her race. Much to everybody's dismay, Leslie wins every race. Jess is not happy with her, and he continues to be mad at her through most of the week.
His attitude starts to change in Miss Edmunds's class. Miss Edmunds brings joy to Jess no matter what. He has a bit of a crush on her, and he can't help but be in a better mood in her class. Miss Edmunds then begins to play on her guitar and lead the class in song. Jess is swept up in the "pure delight" of it all, and he smiles at Leslie. Jess immediately mentally scolds himself, and then he just as quickly decides that is dumb. Jess decides that he is going to have a better attitude about Leslie, and then he smiles again. The moment is a turning point in their relationship.
Caught in the pure delight of it, Jess turned and his eyes met Leslie's. He smiled at her. What the heck? There wasn't any reason he couldn't. What was he scared of anyhow? Lord. Sometimes he acted like the original yellow-bellied sapsucker. He nodded and smiled again. She smiled back. He felt there in the teachers' room that it was the beginning of a new season in his life, and he chose deliberately to make it so.