According to this page -- http://nces.ed.gov/FastFacts/display.asp?id=76 -- the average cost per year at a private, non-profit U.S. college (i.e. not state subsidized) is $39,000. That includes tuition plus room and board.
Let's compare that to Germany, just to pick one European country as an example. According to this page -- http://www.marketplace.org/topics/education/learning-curve/how-german-higher-education-controls-costs -- the average cost of a year at a German college is $32,000. That's what the government spends: it's free to the student.
This comparison is not entirely fair as the number for Germany does not include room and board, and even private U.S. schools get government money in various ways that is not included here. But from this admittedly very simplistic comparison, it appears that the true cost of education in the U.S. and Germany is about the same, at least in the same ballpark. Of course you could debate the quality of the education received endlessly: that's not easy to measure.
Where does the money come from in Germany? From taxpayers. Where does the money come from in the US? From a mix of the student and his family, government assistance, and private scholarships. Most American students borrow most of the money and pay it back after they graduate.
So higher education in Germany is free in the sense that the student doesn't have to pay at the time he attends. But he ends up paying for it through his tax dollars for the rest of his life. As the total cost is about the same, he's going to end up paying about the same amount in extra taxes as the American spends in tuition.
There's less practical difference between the two systems than you might at first think. In Germany students pay nothing while attending school, but then pay for it through taxes for the rest of their lives. Maybe 40 years from graduation to retirement? In America most students pay little or nothing while attending school, but then pay for it through student loan payments for an average of about 20 years.
Poverty doesn't keep a German out of school because it's paid for by the government. But low income Americans can get all sorts of financial assistance and then get loans to pay the rest, so poverty isn't that much of a bar to education in the U.S. either.
You could debate the pros and cons endlessly. The German spreads his payments out over his entire life while the American concentrates it into 20 years. The American may find himself unable to make his debt payments if he can't find a good job, etc, while the German's taxes are presumably based on his income so while he may complain it should still be manageable. The American can decide how much he is willing to spend for college, while the German cannot decide how much he will pay in taxes to support education. In Germany if you can't pass the entrance requirements, you're out of luck. In the U.S., if you can't pass entrance requirements you can't get into your first choice school, but you can almost always get in somewhere. A German who isn't admitted into college still has to pay taxes for others to attend, while an American who doesn't attend college doesn't have to pay for it. (Well, he still has to pay taxes to support the various government programs that exist, but these are much less than what the German pays.) Etc. I'm sure you could think of other pros and cons.