In general, beating a rival and/or winning a league championship in a popular sport leads to higher alumni engagement. In addition to being good for the overall social health of the alumni community, it also leads to higher donations.

At Harvard specifically, beating Yale in football is a big deal within the community.

At large state schools, sports can function as an incredible arm of marketing. In addition to benefits for the alumni community and donations, it also can impact recruiting to undergraduate and graduate programs. Case in point, check out the change in the quality of the student body at the University of Alabama since Saban has been coaching the football team. It has improved not just the profile of the undergrad student body, but also the profile of the law school student body (among other grad schools, I’ve been told).

I will also add that being a varsity athlete puts the student in a totally-not-a-club club that makes a difference in certain social circles (especially in the northeast corridor).

It's also about making lots of money, right? I don't blame the university, though. American culture obsesses over sports. In how many towns the biggest annual event would be the baseball game of the local high schools? In how many schools members in the school's sports team are like superstars? Still remember that Caltech was mocked because they finally won a collegiate basketball game in 37 years. It's still beyond me why basketball game matters at all for a college.

I think athletic success increases the awareness of the college among potential applicants, more people apply, the selectivity percentage goes down, the college looks more selective from a naive perspective used by college ranking formulas, its ranking increases, and thus its prestige increases.

I think it depends on the school how important this is, I kind of doubt Harvard needs to raise awareness. But there are other benefits of sports that will also depend on the particular school.

One factor that comes to mind is location- Notre Dame is in the middle of damn nowhere, if they did not have the appeal of the football team I think they would have a harder time attracting students to actually go there (regardless of name recognition).

It also will probably depend on how much the rich alums care about sports, so again for a school like Notre Dame that has always been big on sports and is also very big on legacies and promoting alum community and what not, it makes sense.

To be honest I don't see why it matters much for today's Harvard though, at least none of the benefits I can think of really line up. Plus Harvard can never attract the truly best athletes because no Ivy scholarships/the generally lower level of Ivy competition in most sports. Maybe they just care about being better than Yale.

It's a high-variance form of marketing.

Well, any student who's interested in playing sports at school will likely be interested in that sort of thing.

I think more generally it adds to the general culture of the school - people will pick up on the fact that some athlete they know went there and think that's cool, even if it doesnt impact their academics at all.

More cynically, lots of students love to associate their school's legacy with their own ego

The kind of upper-class people who actually run Harvard, and especially who run the feeder prep-schools for Harvard, have a really weird fixation on sports. Seriously.

Sport games form bonds, in other words they make networks.

They also bring back older graduates for a piece of nostalgia and bit of donation.