When Leslie and her dad first start working on renovations to their house, Jesse feels awkward and uncomfortable around Bill (Leslie's dad, otherwise known as Mr. Burke) and even feels a kind of vague dislike toward him.
The main source of all these negative feelings toward Mr. Burke is that he's taking up Leslie's time, making Jesse feel jealous. Jesse would much rather be playing with Leslie, having fun in Terabithia together, but instead she's busy with her dad. For that reason, Jesse resents Mr. Burke. We can understand his feelings: we'd also be annoyed if someone took away our best friend, resulting in our own boredom and loneliness.
Another reason for Jesse's awkward, slightly angry feelings toward Bill is that he can't understand why Leslie is so interested in spending quality time with and getting to know her dad. It's sad to realize this, but Jesse barely has a relationship with his own dad, so he doesn't understand it's normal to hang out with your parents and learn more about them. Deep down, then, Jesse may be feeling jealous of the mutual affection between Leslie and her dad. Jesse would love to have more approval and attention from his own dad; since he doesn't, it makes him vaguely unhappy to see Leslie bonding so well with her dad.
Happily, the tension between Jesse and Leslie's dad resolves itself when the two men get to know each other as Jesse joins the Burkes to work on the home renovation projects with them. It's the perfect solution: Jesse gets to hang out with his best friend Leslie again, and Jesse also gets to know Leslie's dad well enough that he no longer feels angry or awkward around him.