Summer classes would usually just be for undergraduate students and perhaps a few for master's degree students, but not usually at the doctoral level - especially if he hasn't started his program yet.
I think BK just arrived in Pullman in late June or July to get settled in to his apartment, get his ID card, get his university email set up, parking pass, buy supplies, check out the library, the university bookstore, buy the books assigned to his fall seminars, and overall just find his way around campus and the community. If his new TA departmental office had already been assigned, he could stop by the department and have the staff give him a key and help him get settled in there, also.
Very normal to arrive early for an incoming graduate student, especially if they have moved across the country to start a doctoral program. There would be a lot to do before the start of the semester.