It’s partly a “business play,” but there’s more to it than that. The University of Michigan is deeply committed to excellence in all regards, and to academic excellence in particular. It’s just deep in the university’s DNA, and it pervades the culture. They know they have a duty to the people of Michigan, but they see their unique role in that regard as providing the best possible educational experience to the state’s top students, in part by bringing in the strongest possible entering class. Going back generations, they’ve done that in part by admitting each year a very strong cohort of OOS and international students. That approach also adds geographic diversity and a diversity of perspectives to the student body, and that’s a hugely important part of the learning experience for Michigan residents.
That approach might not work so well if Michigan State wasn’t standing there 65 miles away, a fine alternative for any in-state applicant who might prefer Michigan but isn’t admitted—as well as the destination of choice for many, including some of the state;'s top students. To that extent, Michigan needs Michigan State. And the relationship is symbiotic; it also helps Michigan State by giving them an in-state rival they can excoriate as “arrogant” and “elitist,” which certainly fuels their athletic teams and fans, but also becomes a selling point in recruiting in-state students.