Three times, the Torah tells us not to eat a kid (a baby goat, not a child!) which has been boiled in it's mother's milk. [An important aside: There are some tricky translation issues here. "Milk" may actually be "fat" or, more precisely, one particular kind of fat. "Boil" may be "seeth," which is different. In the end, these issues probably don't matter, because regardless of what the original language may have been, the law developed into the concrete system which we have today]. The Rabbis of old understood this to be a general injunction to not eat an animal with any milk from its mother.
Very quickly, this law expanded - just to be safe (to prevent accidental mixing), we shouldn't eat meat with the milk of any animal of that species, not just the mother (how can you be sure which animal this milk came from?), and then, for the same reasons, to any milk from any animal. Even chicken (and other birds), which don't have milk, came under this restriction (although, there was much more debate about this in Talmudic times - roughly 2000 years ago), because of the ease of confusion (if you'll eat one meat with milk, it's easy to accidentally eat another meat with milk, too).
This is all a classic case of "building a fence around the Torah." We enact extra laws and regulations to make sure that we don't even come close to violating important laws. Eventually, the laws expand so much that the sages of old referred to the laws of kashrut as a mountain suspended by a thread - a huge amount of law, based on the tiniest bit of textual grounding.
My own opinion is that the origin of these laws is not what's really important. Whereever these laws came from, they have evolved over the centuries into a distinctively Jewish way of eating. I don't avoid mixing meat and milk because the Torah says, "don't boil a kid in it's mother's milk" per se. I avoid mixing meat and milk because that's a Jewish dietary practice. I want to eat like a Jew, and this is one of the ways in which I do so!
Answered by: Rabbi Jason Rosenberg (Emeritus)