This is a rather odd idea that's been a bit tough for me to pin down solidly.
Basically, what we've got here is the application of human morality to animals. It's classic animism, applying traits that we know and are comfortable with to animals. Bunnies are cute and cuddly. Dogs are loyal. Cats are aloof. Dolphins are happy and intelligent. It's a way for us to try to understand animals, bring them to our level rather than try to understand the level they exist on.
So we apply human morality to cows and chickens and doggies and squirrels. You might ask, why is that so bad? Well, the biggest reason for me is that granting animals similar liberties to humans just drags down on our own concepts of freedoms and liberties. Instead of elavating the rights of the animals, it much more drags down on our own rights, debasing them and minimalizing them.
I know that I am entitled to my life. I choose to lead it as I do, and I give that decision to no other person. It is my basic right as a human being from which all other rights stem. I hold it holy; it is one of the only things I hold holy. When that right (or one similar to it) is extended towards cows or dogs or anything not-human, it degrades the sense of importance associated with that right to my own life. Of course, that degradation isn't true in fact, but getting people to believe it isn't so important or special is the first step to getting them to surrender it.
If you want to talk about biology, how we're designed, how we're 'meant' to be, know this much. We are killer-apes. That is the first reason (the only reason?) our race developed intelligence: to become better at hunting and surviving and being the baddest, most dangerous, most adaptive creature on this planet. Survival of the fittest; human beings are a direct product of it. It might be argued that we can 'elevate' ourselves above that level (I would argue against that, if only to be devil's advocate), but let's not make-believe about what we are 'intended' to be. Our ancestors were amazingly efficient and successful killer-apes. 'Nuf said.
Myself, I value human life above animals. Using mice to research medicine? Hell, they're just starting to treat spinal cord injuries in mice; I'm willing to bet that Christopher Reeves will walk within the decade, as long as gov't and protesters mostly stay out of the scientists' way. Food? I'll be honest, bacon tastes good. I know full well where it comes from, and I still like ham and beef and (gasp!) even sausages of unspecified origin. It tastes good, and (at least for me) is much better at fueling me than any fruits and vegetables. Trust me, I tried a veggie diet for a while a few years ago. I just felt run-down, without energy, and caught 2 colds (I never get sick!) before I gave up and sat down for a steak. I don't know if it's real or psychosomatic. That's just the way it is, for me, and for a lot of other people.
Look, the main thrust here is that granting animals rights only demeans the ones we have ourselves. There are a lot of other arguments here, but that is the main one I would stand by and die defending.