Not quite. It's a circular system. Money goes places like university tuition, housing, and dining.
"To the students" means "to pay the university."
I've walked through the complex and bizarre accounting, but the short story is:
* The university wants "general" funds, which it can use to pay million-dollar salaries and for fancy dining establishments, rather than restricted funds.
* Almost all money comes in as restricted funds (e.g. for teaching, research, or otherwise supporting the mission of the school).
* There is a complex network of funnels to go from restricted funds to general.
For example, tuition is artificially set high. It's then discounted to anyone who can't afford it. However, government research grants, fellowships, etc. pay the full tab. If the cost to educate a student is $20k, the university will have a sticker price of $50k. For deep pockets sources, it will take $20k to educate the student, and funnel $30k into general funds. To make donors feel better, it will then advertise it actually costs $200k to educate that student (!).
With research, the channels are "overhead," which is often set at around 60% of elite schools.
These are well-polished channels. Any money coming in for students, you can assume is ultimately landing in someone else's pocket. It's just passing through tuition, housing, or similar along the way, the price of which is always decoupled from reality (in both directions, actually, since it's just a financial game).