Even though I'm not Chaim, I thought I'd chime in here.
It comes from one of the many laws outlined, I believe in the book of Numbers (
) where it said "you shall not seethe a kid in its mother's milk. From what I understand, in Judaism, milk symbolizes life, since the animal consumes it to grow big and strong, whereas blood symbolizes death. We don't mix the two.
Also along with that law are a bunch of others relating to the slaughter of animals, which must be done as humanely as possible. For instance, we are not allowed to slaughter an animal in front of its mother, or on the same day as its mother. Furthermore, if an animal you are trying to slaughter escapes, it's against the law to chase after it.
Hope that kind of answers your question.
Now on another note, does anyone here have a link to the audio of Rabbi Tovia singer discussing his first experience at an animal slaughterhouse? I think it would be good to share here.