In the Torah the rabbis have identified two kinds of laws, hukim and mishpatim. While mishpatim have explicit rationales in the Torah, hukim do not. Kashrut, falling into the latter category, has been the subject of a great deal of commentary aimed at helping us to understand it's purpose and spiritual benefit. The Torah does link kashrut to kedusha, or holiness, and connects the observance of kashrut to a concept often referred to as imatatio dei -- imitating God. This helps us to understand that these dietary laws were instituted on some level to help us to fulfill our roles as partners with God in a more complete way and to encourage us to behave in our lives in a more Godly fashion. Specific values, such as tzar baalei hayim, the ban on cruelty to animals, as well as idea of spiritual discipline and curbing our animal appetites have also been applied to these laws. Most rabbis and scholars, therefore, see the observance of the laws of kashrut as opportunities for us to spiritualize both the preparation and consuming of food - a daily activities that necessarily find themselves at the center of a human being's life.
Specifically in regard to the separation of milk and meat, the Torah states three times, "You shall not cook a calf in its mother's milk" (Ex. 23:19; Ex. 34:26; Deut. 14:21). Our early Sages, and the rabbis who followed after, did not see this verse only in its literal meaning. Rather, perhaps due to its three-fold repetition, they gave it a more expansive interpretation, seeing it as the source for the general prohibition of eating milk and meat products together. Later, to avoid confusion, the rabbis extended this prohibition to include the mixing of fowl and dairy products, as well.
The spiritual rationale for this particular law that is most commonly applied is that we must not eat "the flesh of the animal together with the milk that was meant to sustain it" (Kushner, Etz Hayim). In line with the concept of respecting the animals that supply us with food, we find it cruel, spiritually and metaphorically, to cook and eat the flesh of the animal in the very life-giving substance meant to nourish it. Therefore,this general prohibition is supposed to develop in each of us a respect for life, an appreciation for God's creatures as well as a sensitivity to any type of cruelty. In addition, Maimonides saw this particular law as teaching us a more spiritual lesson. He claimed that kashrut in general, including the prohibition of eating milk and meat together, was en exercise in curbing our animalistic appetites and instilling in us spiritual discipline. The most base element in us might want to eat anything we wanted at any time. However the sacred dimension of being human requires that we recognize that we must be humble in our world outlook and recognize the limitations that we must place on our own personal desires. Kashrut helps us foster that outlook and develop that spiritual discipline. It aids us in recognizing God's blessings and in becoming more appreciative for that which we have.
There are three categories of Mitzvahs: Civil ordnances, commemorative and statutes. Civil ordinances are those mitzvahs which all civilizations have adopted - stealing, murder, etc. Commemorative mitzvahs mark an event - Shabbos, the creation, shma, tefillin, tsitsits, the holidays celebrate the exodus from Egypt, the third category is by far the smallest. These commandments came with no logical explanation. They include the laws of ritual purity, mixing wool and linen and the laws of keeping kosher.
Keeping kosher is not a health code even though there may be health benefits. The Jewish body is considered holy by G'd and needs a higher grade fuel to keep it running. Animals eaten do not kill their prey or eat carrion. They are domesticated. Shell fish are scavengers, eating all the waste of other sea creatures. Milk and meat has no imaginable reason. We keep it as an exercise in trusting in the Almighty's commandments.
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!