As a simple example, a school with a division 1 football team like Ohio State will spend approximately $100 million per year on athletics but that will be less than the athletics programs bring in. The mens football and basketball teams will bring in almost all of the revenue, and will support all other athletic programs (and athletic scholarships) at the university. The Ohio State football stadium seats more than 100K, and the minimum ticket price is $22 so you are looking at more than $2M in gate revenue for games, plus TV contract revenue, plus merchandise it sells all over the state, plus donors and alums who will pay far more than $22/seat for corporate boxes or to attach their name to some aspect of the program for commercial purposes.
Stadiums can sometimes look for public subsidy when being replaced or re-built, but they can equally make a claim that their presence brings in millions of dollars of revenue to the local area over the course of a season and that the cost of the stadium can easily be amortized over decades (something that is actually easier for a collegiate program in a mid-sized city to claim than a pro team in a major urban center.) The ancillary facilities like practice space, offices for coaching staff and other infrastructure are almost always paid for by the athletic programs to the best of my knowledge.
Stadiums can sometimes look for public subsidy when being replaced or re-built, but they can equally make a claim that their presence brings in millions of dollars of revenue to the local area over the course of a season and that the cost of the stadium can easily be amortized over decades (something that is actually easier for a collegiate program in a mid-sized city to claim than a pro team in a major urban center.) The ancillary facilities like practice space, offices for coaching staff and other infrastructure are almost always paid for by the athletic programs to the best of my knowledge.