Brinker has started to wonder if Gene isn't hiding something about the incident in which Finny fell from the tree and shattered his leg. As Gene states, describing Brinker's character:
Before he had always known and done whatever occurred to him because he was certain that whatever occurred to him was right.
When Finny and Gene arrive at the "court," not knowing what is about to happen, Brinker verbally jumps on Finny, saying to him:
We don’t want any mysteries or any stray rumors and suspicions left, in the air at the end of the year, do we?
In other words, Brinker suspects—and therefore is sure—there was foul play with the shaking branch that caused Finny to fall, and he wants to expose the truth. He is insensitive to how disturbing his method of doing so is to Finny and Gene.
Gene also realizes that Brinker is "enjoying" this trial, because he likes the idea of doling out justice.
Brinker is particularly insensitive and uncaring about the fact that his surprise courtroom hearing is very uncomfortable for both Gene and Finny. He is the type of person who enjoys the power of ferreting out the truth.