As CTO said years ago, our applications from new students increased by 25 or 30% after we won the championship. That allowed the University to be more selective in who it decided to admit, which drove up the rankings.
People who are proud of their school and who have been successful in business also tend to donate more money to their school. Sporting events continue to provide networking opportunities to graduates, and bigger events draw bigger crowds of enthusiastic alumni.
Sports and academics are not as separate as you think. The University is still a business, and they have to sell "quality of life" things to get kids (or their parents) to pay $70,000 a year for a mostly worthless Bachelor's Degree.