Why would a professor write many recommendation letters? A practical question: 
I have now read a few times that to get a PhD position in the US you need 3 reference letters.
I assume most people apply for more than one position, so you need those 3 letters for every position.
Also, there are a lot of applicants.

So how does one get a professor to write more than one letter? 
If I apply for say, 5 positions, I would need 15 different letters, 5 per person if I am always referring to the same persons, even if not probably more than one letter per person---and I am probably not that person's only student who wants letters.

I cannot possibly imagine professors in Europe going along with this (which is why it is becoming common here to just give the address of the referee, not send in a complete letter), so how is it in the US?

Do US students only apply for one or two positions and simply abandon the thought of grad school if they do not get it, or do they really get their professors to write that many letters?
The answer is simple: you generally only write one letter per person and only customize it insofar as changing the name of the institution to which it is sent (if even that: "Dear Admissions Committee" works pretty well). If there are very different types of programs involved, you might customize a bit more, but not much. That means the incremental work needed for many letters is not much more than is necessary for one.

If you think about it, it should make sense: a recommendation letter is primarily about the candidate, not the institution. If a recommender writes the best letter they can for one institution, how much could it vary and still be the best letter for another?