This is similar to the sentiments of Homer Simpson — “If we weren’t supposed to eat animals, why are they made out of meat?”..
The case for animal rights is based on utilitarianism — the general idea is that animals feel pain, so they ought to have the same kind of moral consideration as all other beings that feel pain. Thus, we ought not eat them.
It’s a bit more complex than that, but that is pretty much what it comes down to as a justification for vegetarians.
It becomes a bit more complex when we add the idea that animals ought not be used for medical testing. In that case, it’s hard to support a ban on medical experiments based solely on utilitarian grounds. Think about it this way — if all beings that feel pain are equally worthy of moral consideration, it ought to be the case that some beings can be sacrificed in order to increase the general welfare of all beings. Medical tests develop treatments for beings that would feel pain (or die) without the treatments, so the animals who had to be sacrificed in order to develop those treatments are contributing to a project that should be supported on utilitarian grounds.
Animal rights folks (Singer in particular) argue that animals are not sufficiently similar to humans, so the results of animal testing of medical procedures yield results that are not worth the sacrifices on the part of the animals. This seems to be a weak argument, because the results of scientific tests are open to interpretation and the interpretation is done by scientists who see an analogy between the system of the animal and the system of the person.
Because I’m generally not a utilitarian, I can reject animal rights on that ground alone. It seems to me that I have a set of intellectual skills that my cats don’t have. In particular, I can assign value and make long term decisions about my behavior in a moral context — and my cats rely on instinct more than anything else. So, while I have duties to my animals, they don’t have rights.
Additionally, I have duties to my cats because I assumed those duties when I adopted them.