How does Jess become acquainted with Leslie's father in chapter 7 of Bridge to Terabithia?

Jess gets to know Leslie's father by volunteering to help fix up their house.  By working alongside him and spending time with him and Leslie, Jess gets to know what Bill Burke is really like.

Leslie had been spending a lot of time working on the house with her father, and Jess had actually been a little jealous, because she did not seem to have time for him anymore.  Jess, who has a very different relationship with his own parents, does not get it when Leslie "glowingly" tells him how she is "learning...to understand her father"; to him, parents are no more "meant to be understood...than the safe at the Millsburg First National (is) sitting around begging him to crack it".  Leslie finally confronts Jess about his feelings, and suggests that Jess come and help at the house and get to know her father.

At first Jess is uncomfortable being around "Bill", but he soon discovers that the man is a highly intelligent and genuine person.  Jess is used to working hard, and is skilled in a number of ways that help out with the home improvement project.  It pleases Jess to find that despite Bill's deep capacity for book knowledge, there are ways in which Jess can be "really useful to him, not a nuisance to be tolerated or set out on the porch like P.T".  Bill is truly thankful for Jess' help with the house and honestly appreciates him as a person.  By taking the chance to get to know Leslie's father, Jess' horizons are expanded and he finds that there are families which are far different from his own (Chapter 7).

What are the differences between the fathers of Jess and Leslie in Bridge to Terabithia?

The first thing to point out is that Leslie’s dad is far more developed as a character than Jess’s dad is. While we never even learn the first name of Jess’s father, who is referred to simply as Mr. Aarons, we know that Leslie’s father is called Bill.

Mr. Aarons has old-fashioned ideas about gender roles, and Jess is given far more work to do around the farm than his sisters purely because he is a boy. We are given the impression that Mr. Aarons may love his daughters more than he loves Jess. Bill Burke, on the other hand, seems to be a particularly progressive parent. Leslie addresses him and his wife by their first names, and they seem to allow her to make her own decisions about how she expresses herself.

Another poignant difference comes to light when Jess starts working on the old Perkins place with the Burkes. When he is able to do things that Leslie’s father cannot, he receives genuine praise and is made to feel like he is making a difference. It is apparent that Jess has not felt these sentiments at home from his own father. Bill Burke thus appears to be far more approachable than Mr. Aarons.

While Mr. Aarons and his wife adhere to certain traditions, like an annual trip to church for Easter, Bill Burke and his wife have never taken their daughter to church.

Ultimately, when tragedy strikes, it is apparent that the two fathers have one thing in common: they care deeply for their children, even though they have different ways of expressing themselves.