In a lot of the EU, students from other EU countries don't have to pay tuition fees. Foreign students? Not so lucky, and language doesn't matter. If you aren't from the EU you pay increased fees. For example in Sweden you pay about 15,000 EUR/year for a science degree. In terms of language, you have to already demonstrate a proficiency in English and Swedish just to be able to get in.
Also all of this assumes you can get a visa and get admitted. People from other EU nations, no problem, you can live and work anywhere in the EU, that is a big part of what the treaty means. Non-EU individuals have to get a student visa, the requirements of which vary.
And of course none of that deals with the cost of food, housing, transport, etc. You are on your own for that, barring a scholarship.
This is a subject I have more than a passing familiarity with, as my sister is currently working on her PhD in Europe (at two universities, one in the EU one outside of it). She got a generous grant that pays all her tuition, living expenses, and even some extra but that isn't what all students get. It wasn't as though she just walked in and said "I'd like to go to school here," and they said "Certainly, please come for free!"
Also she even had an easier in than many: She and I hold Canadian citizenships. Canada is a commonwealth country and England is in the EU so that makes a lot of the visa shit way easier than it would be for an American, not that it wasn't still a big production.
It is exceedingly narrow-minded to suggest that an American should just "Emigrate to an actual civilized country instead of a pretend one," for school, as though such a thing were trivial to do and people only did not out of ignorance (not to mention the misplaced cultural supremacy of the statement). No, it turns out that you can't just graduate from an American highschool and say "Well screw the US, I'm off to Europe!" and walk in and go to school for free.