How do academic grants work in the U.S.? When a professor is awarded a grant, what happens next?

Does the professor get the money?

Does the school get the money? If so, why?

Is it extra income for the professor?

How about when there are multiple professors on a project? Is the money theirs? Do they split the money?

How about when a graduate student gets funding from an agency? Does the student get the money? Does the school get it?
This answer is for the United States, though I believe that it works in a similar way in Europe.

The money goes to their university, not to them.  The terms of the grant say what the money is allowed to be used for.  The university either pays these expenses themselves or reimburses the professor if they pay out of their pocket.  Everything is documented by things like receipts, and universities periodically get audited by granting agencies.

Sometimes the grant includes money for salary (in mathematics, it is common for NSF grants to include 2 months of summer salary), which is dispensed by the university as part of the professor's paycheck.

In mathematics, the typical things that a grant pays for in addition to salary include travel, books, and graduate students, though often there are other things as well (for instance, my current grant has money for a summer program for high school students I run).