First off, we have to define what we mean by "animal rights." Most proponents don't really have a firm grasp on this beyond a vague notion that animals should be respected as people are. While there are some that think that chimps should have voting rights (in spite of their inability to read a ballot) and all sorts of other insane notions, most animal rights activists stick to an idea that animals should be free to do as they please. In this view the animals should be as free as people to roam about and do their animal things unmolested and restrained by humans. After all, animals are organic creatures just like people and should therefore enjoy the same rights...
The problem here is that rights are not based on biology. If they were, we'd have to apply the equal protection clause of the Constitution to every petri dish full of virus cultures. After all, they are based on DNA just as we are, so they must have the same right to go about their viral business. This is on its face rather ridiculous; a virus is nothing but a protein coat surrounding a DNA core. It can't even reproduce without forcefully taking over the machinery of a host cell.
I have coined a term to describe what rights truly are based on, which is "reciprocal responsibility." What does that mean? Reciprocal responsibility means that you, as a holder of a particular right, are held responsible for respecting the similar and equal rights of others. You have a right to free speech, but you must respect both another's right to free speech, and must not misuse your right of free speech against another (in slander, for example). You have a right to keep and bear arms, but you must not infringe on another's right to bear arms, nor shall you use your arms in unjustified force against another. This is reciprocal responsibility. A violation of reciprocal responsibility opens you to criminal and civil liability. No right exists without this attending responsibility.
The problem with animal rights is that animals are incapable of recognizing and respecting reciprocal responsibility. If left completely free, animals would habitually engage in acts that would be criminal if performed by human beings. Bears and cougars would attack (assault) people. Dogs would defecate on (vandalize) people's lawns. A million other "criminal acts" would be perpetrated by animals, in violation of others' rights. This leads to one of two outcomes:
1) Humans and animals would necessarily have to have different standards of behavior. Acts against persons or property that would be criminal for humans, would be allowed for animals. In other words, animals would have superior rights to humans. We'd be constantly cleaning up after the criminality of animals, and would effectively be enslaved to creatures that didn't have a second thought to us and our rights (which undoubtably do exist). This is backward, to put it mildly.
2) To protect us from the criminal acts of animals, we'd have to imprison animals that "break the law." Since animals don't have the reasoning ability to make rational choices, most animals would eventually end up imprisoned for breaking the law. So we'd be locking up all these "freed" animals...how exactly does that respect their "rights?"
The concept of "animal rights" is an enviro-fantasy based on a misguided and ill-considered desire for animals to be happy. We have a responsibility to animals, not to mistreat them or cause them undue pain or suffering. There are, and should be, laws against animal cruelty. But to ascribe "rights" to animals goes against the concept of reciprocal responsibility which is inherent in the very definition of rights. Those who cannot respect rights cannot hold them...ergo, no rights for animals. Sorry PETA.