The school case and a criminal or civil case are independent of each other. The school receives the complaint and, in the case of a potentially criminal matter, refers it to the police for further action - that aspect of the case then becomes the responsibility of those officials to decide if they pursue an investigation/prosecution, etc. I've seen quite a few of those cases where the criminal investigation is a nonstarter because the victim refuses to make a criminal complaint.
The school however, is still obligated to pursue the case as a potential violation of student conduct policy, which is what apparently been done and decided in this Yale case. Each school has a set of procedures that govern the evaluation and penalty, but on the schools part, the strongest action they have available is to expel the student. No matter the outcome, FERPA laws are still in play, and there won't be an official description of the case unless the students agree to the details being public. That looks like what happened here - the school ran the case, found him in violation of student conduct and expelled him. They can't say why, and he isn't saying anything about it either, so who knows what he did. As I noted, my guess is that, if this was a sexual assault case, the victim wants him gone from the school, and is keeping the possibility of a criminal charge as leverage to keep him quiet.