Anyone who lives with a cat or a dog knows that animals not only have personalities, but that they also have memories, fears, wants, and desires. The dogs and cat that I live with seek out comfort, avoid pain, and desire companionship, and it is clear to me that they suffer as acutely as I do (if not more acutely) when they are hurt, or sick, or scared. Moreover, having spent time around animals ultimately destined for slaughter, I also know that pigs and cows and chickens are also capable of these same pleasures and pains, and what appears to be a subjective awareness of their surroundings, yet for reasons that no one can really justify, we snuggle up with one set of animals called "pets," while we eat another set of animals called "livestock."
When it comes down to it, the case for animal rights is really a case for adopting a thorough moral and ethical stance in favor of treating like cases alike. My own outlook has been shaped by the ethical theory of Gary L. Francione, who argues that though animals and humans are clearly different, they are alike in the sense that they both suffer, and are both sentient. For this reason, Francione argues, animals should receive equal moral consideration. Most importantly, this would mean extending to animals inherent value, or really bringing them into the moral community by recognizing that certain aspects of their personhood cannot be "sold away" or sacrificed for the benefit of another. Put most simply, because animals are like us in some relevant regards, they should be treated like we would be treated in those instances.