Here's how it was in 1973, when I was appointed to the U.S. Naval Academy by Senator Frank Church.
Admission to the military academies is by appointment, anyone interested in going to one of them must apply to their Senators or Congressperson, who each are allocated one appointment. A couple of other ways in are if you are the child of a serviceperson who received certain high honors (don't remember what they all are offhand, though one is to be dead from war), Presidential appointments, or, if you are already enlisted, you can petition your commanding officer for a nomination.
Appointees become employed by the United States and are paid 1/2 the salary of the lowest commissioned officer, get room and board, and the four year education is "free" (it is not a liberal arts education...). The deal is that you promise to serve five years after graduation (you can resign during the first two years and avoid the five-year commitment, however, you have to pay back some of the costs). I quickly learned I had made a serious life-mistake and processed out at three months.
I'm waiting for someone to fact check Carson's claim that West Point says "four year scholarship" anywhere on its website. That would constitute a lie about a lie (speaking of the Tar-Baby).
