"Why do people read books?" — "Because people read books to get information."

Solution 1:

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."

Solution 2:

The question is: "Why do people read books?" The answer given is: "Because people read books to get information." If that answered the question, the following sentence would be true: "People read books because people read books to get information."

Consider:

"Why should I trust you?"

• "Because I'm honest."
  
• "You should trust me because I'm honest."
  
• "Because I'm honest, you should trust me."
  

All three of these are perfectly acceptable answers. The last two are complete thoughts that contain within them the answer to the question. The first is a sentence fragment, but is acceptable in spoken English because it contains only the answer, essentially completing the question.

But look at: "Because people read books to get information."

It's not a complete thought. It's a fragment. And it doesn't meet the informal spoken exception of giving just the answer to the question. So essentially, it's a sentence fragment that doesn't conform to any informal spoken English pattern that allows them.

A native English speaker would likely see it as disfluent.

Solution 3:

"Why do people read books?"

"Because people read books to get information."

Why does the answer read a bit awkward? Because it starts with a conjunction.

Strike the first word from the answer, and it reads fine:

People read books to get information.

Because is often used as a leading word when answering a why question, so it's easy to see why a child would start with that word. Perhaps a comma might help it read better:

Because, people read books to get information.

I'm not saying the comma is required. However, we often see a comma used when a sentence begins with a leading conjunction.

Sometimes the central issue revolves around words being awkward in a given context. That said, the fact that a sentence can be improved doesn't necessarily mean it's grammatically wrong.