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.