As a foreign national? As, say, a US citizen without a work visa? Does that really work?

The idea that a foreign country would subsidize one's college education is pretty alien to a US citizen. Just going to a state university in a different state costs extra money in the USA.

Or are you just asking: Why would people who are already lucky enough to have some kind of "old European" citizenship ever want to study in the USA instead? I'll let those folks answer that question, since I don't really know.