Brinker is unwilling to accept the mystery (or the unknown) that surrounds Finny's fall from the tree. Though Finny has already said that he must have simply slipped, Brinker does not believe that this is the real explanation.
To get Finny and Gene to attend the trial, Brinker essentially arrests them, bringing several people with him to Finny and Gene's room. Brinker does not use force but threatens to do so.
There is reason to question Brinker's motivation in this episode. One potential explanation is that Brinker is stressed by the approach of the war with all of its unknowns and its prospects of danger. He therefore wants to assert his ability to control events as much as he can and also eliminate the unknown as it exists within the scope of his social world at Devon.
These motivations are not articulated in the text itself. Instead, Brinker attempts to suggest that it is in Gene's interests to investigate the incident of Finny falling from the tree.
Then, insinuating that Gene has a "personal stake" in finding out the truth, Brinker suggests that there are a lot of unanswered questions about Finny's accident, and that the sooner that these questions are brought out in the open and resolved, the better it will be for everyone.
There isn't really much mystery here in this portion of A Separate Peace, I'm afraid. Brinker sends a couple of guys to escort them to the assembly hall for a late-night, forbidden adventure, and there you have it. What's more important, I think, is the apparent tone of the incident at the beginning. It's pretty clear Brinker had no real sense that the night would turn out like it did, Finny's fall or not. It's called a mock trial because it's not meant to be serious. In other words, it may look like a trial and sound like a trial, but it's not really a trial. However, when Finny's suspicions and desire to finally hear the truth meet Gene's guilty conscience, this is bound to be more than a mock trial.
In A Separate Peace, "Brinker and three cohorts" came with a lot of commotion into their room at 10:05 and "half-lifted" Gene and Finny out the door. They hustle them down the stairs, then direct them to the mock-trial.
Better than my opinion, I can give you Gene's opinion as to why Brinker is so intent on investigating Finny's accident:
I had no idea what Brinker might say or do. Before he had always known and done whatever occurred to him because he was certain that whatever occurred to him was right.
Brinker thinks he knows something about Finny's accident, and he thinks he's right. And he wants to prove it. Whether or not he has ulterior motives--such as wanting to expose Gene and see him put down--I don't know. That would be speculation and guess work, unless some other textual evidence exists that I don't remember. Which is possible, of course.