もらっていい（ですか） is a pretty common way of asking for something, and I would translate it either as "can I keep this" or "can I have this?" As you said, もらう is receive, so it's not hard to see how this actually works out.
There are several ways to ask for something, each with its own nuances, but using もらっていい will generally mean an explicit change of ownership. If you want to make it more polite, いただけますか will also have the same sort of connotation.
On the other hand, if you want to ask for something without keeping it, you can use a variety of other verbs directly relating to what you want to do, like その本を見ていいですか？ or 使っていい？ If you want to keep it general, you can say, for example, この本を貸してくれますか？ or 借りていい？, meaning "would you lend" and "can I borrow" respectively.