That's a tricky question. Start with a different, and probably easier (for many people) question: where do humans get rights? There are many possible answers. Some people think our rights come from God (or gods), others take different approaches. Ultimately it's just a matter of deciding what is important and "true". You have to invent your own axioms in some sense, I think. So, once you've done that, you can use those axioms to decide whether or not animals have rights.