Learn Russian: Любой человек может делать то, что он хочет.

Любой means any in the sense of a free choice: it does not matter which person you pick, the statement applies. If we used Каждый, it would mean every person or each person, focusing on all members of a group rather than the absolute freedom of selecting anyone.

Why does the sentence use то, что instead of just что for the English word what?

In English, we just say do what he wants. In Russian, you usually cannot use что alone to do this job. You need the pointer word то (that) in the first half of the sentence to act as the object of делать, followed by a comma, and then что (which) to start the second half. It literally translates to doing that, which he wants.

Why use the imperfective verb делать instead of the perfective сделать?

The sentence describes a broad, general truth rather than one specific, completed action. The imperfective делать is used because this is about generally doing whatever one wants, not finishing a single defined task.

Could I leave out человек and just say Любой может...?

Yes! The word Любой can act like a noun on its own, meaning anyone. So Любой может делать то, что он хочет is perfectly natural Russian and means exactly the same thing.