Translation:How did you get these rare books?
I agree. I thought the sentence meant ‘How did you come by those rare books.’ This was rightly graded as erroneous, but the correct translation looks like my translation was right! There's (almost) no clue that we're actually talking about arriving rather than acquiring.
To figure that out I had to spend quite some time with dictionaries which wasn't made easier by standard combinations like ‘chegar a uma conclusão’.
If this sentence needs to be in the exercise (maybe because it's so confusing) it has to be accompanied by an explanation that is revealed when you check your answer.
Maybe there's a word missing: "Como você chegou CONSEGUIR estes livros raros?"
Yes, that seems better - even if it is not the same language, Spanish has a similar form: "Cómo llegaste a conseguir ..." I wasn't sure, so I left the "a" out
"Como você chegou a esses livros" means "how did you get to those books", "how did you come by those books", "how did you find those books".
If you use "conseguir" alone, "como você conseguiu esses livros":
- How did you manage to find/get those books?
You can't use both together in this sentence with "como". "Chegar a + verb" means "to come to the point of + verbing".
- Você chegou a falar com ela? = Did you manage to talk to her? / Were you able to talk to her?
- O som era tão alto que chegou a quebrar a janela = The sound was so loud that it came to the point of breaking the window. (It even broke the window)
Using both together is used sometimes, but never in questions with como. It will mostly mean "to be able to", but following a process.
- Ele chegou a conseguir falar com ela = He managed to her (at some point)
Your thought on this is also my first thought, how did you get to the source or location. Then I thought from the context, "How did you get to (have possession of) these books. It leaves if open to conjecture. Perhaps "chegou a" should be taught as "get" and drop the "to" from the English.
"Chegar" is very versatile and doesn't just mean "arrive" in the physical sense: http://en.bab.la/dictionary/portuguese-english/chegar-a