It is common to answer a "Why...?" question with "Because..." or indeed another phrase that isn't (overtly) a full sentence.
However, the answer then implies the entire proposition being questioned from the original question.
So e.g. in the following sequence:
"Why do my plants die?"
"Because you don't water them."
the answer implies: "Your plants die because you don't water them".
So in your sequence, the answer would imply:
"People read books because people read books to get information."
Now, the answer is not ungrammatical and could potentially just about express what the speaker really intended to say (e.g. they may have meant "People generally read books because they see other people reading books to get information").
However, it is pragmatically odd and unlikely to express the intended meaning. They probably intended simply the meaning of "People read books to get information". In which case, the appropriate sequence would simply be:
"Why do people read books?"
"To get information."