Great question, and the answer is: There's no real solid answer. Even the rabbis can't figure it out! Some say the priestly garments used wool and linen together so the commandment for the people not to wear clothing woven with wool and linen is because the combination was reserved for holy garments. Other rabbis say the priestly garments were NOT made with wool and linen together! And still other rabbis say the mixing of threads is prohibited because it was part of the practice of the Canaanite heathens of the land (see Leviticus 20:23).
Here are the verses in question:
"Observe my regulations. Don't let your livestock mate with those of another kind, don't sow your field with two different kinds of grain, and don't wear a garment of cloth made with two different kinds of thread." (Leviticus 19:19, CJB)
"You are not to wear clothing woven with two kinds of thread, wool and linen together." (Deuteronomy 22:11, CJB)
It is understood that these verses are both referring to wool and linen and that the prohibition is not a general prohibition of mixing "any two differing types" of thread. (Deuteronomy is not "independent" of Leviticus. The two books document the same commandments given at significantly different times of the exodus, years apart. Deuteronomy was recorded after the original generation had died off. See Deuteronomy 2:15-16).
It seems the greatest body of evidence supports the idea that the prohibition of mixing wool and linen is because YHWH did not want His people doing the disgusting practices of the Canaanites. The Canaanites thought their actions could "influence" their gods, so they often experimented in "mixing" things of value of importance to the "gods" including animals, seeds, and materials. (You can study this via an Internet search.)
In both verses (Leviticus 19:19 and Deuteronomy 22:11), the prohibition is found after a strong admonition to not do any of the disgusting practices of the pagan nations being expelled from the land. See Leviticus 18:24-30 and Deuteronomy 18:9 for example. Also, one can see that Leviticus 19:19 is within the group of verses which immediately follow the opening of Leviticus 19 which pleads for the people to "become kedoshim for I am kadosh, for I am YHWH your God." (Leviticus 19:2). Therefore the commandment to not mix threads is just one example how one becomes kedoshim - holy people. Thus, not wearing clothing woven with wool and linen was one of the "measures" of becoming kedoshim. This supports the idea that mixing wool and linen was a Canaanite practice not to be done by YHWH's people. If the mixing of wool and linen was meant to "please" some false god, then if any of YHWH's people did the same, it would be considered an abomination! One cannot become holy by doing the things of the pagans which were done for "other gods".
Does the prohibition still hold today? Well, yes and no! Yes, it still holds if we have among us pagans who mix wool and linen for influence of their "gods". After all, we would not want to be mistaken as a pagan worshipping a false god. Yes, the prohibition still holds if, for example, you had the misfortune of being given an article of clothing made with wool and linen by a person whose intent was to honor some false god. But "no", if our clothing is simply manufactured with the proper materials to give the clothing strength, durability, fashion, etc., with no intent whatsoever to influence or honor a false god! If you own any garments made with some percentage of wool, and any other material it is okay! It was not made as a pagan attempt to influence a false god, and you are not wearing it for that purpose! If, on the other hand, you are bothered by that blend of wool and whatever the other materials are in the garment; don't know if it was made by a pagan for a false god; or you simply choose to strictly obey the commandment, then don't wear the garment!
The whole controversy of mixing milk and meat had extended the commandment and is tradition. Torah only says don't boil a kid in its mother's milk, which implies showing mercy to animals; and in both places in Torah where it is mentioned it also suggests that all the Mitzvot are to be administered with this same kind of mercy and respect that one would show to a relationship between an animal an its mother. In other words, if we are to show this level of compassion to an animal, then how much more compassion are we to show to one another and to the observance of the Mitzvot?
Concerning the issue of boiling a young animal in its mother's milk - this also was a common Canaanite ritual offering to pagan gods; indigenous pagan magic practiced by pagan priests to increase fertility and productivity. YHWH wanted to set apart and make a distinction between Israel and the pagan nations around them; therefore He prohibited them to imitate these abominable ways. Sacrifices for YHWH were accomplished in a specific way, and often included boiling and consuming the meat of the sacrifice. This whole process was considered "holy", and boiling the sacrifice in its mother's milk, being a pagan practice, could not be permitted as nothing unclean is permitted with a holy process.
The prohibition does not prohibit "mixing meat and dairy!" Eating cheese on a hamburger is perfectly permitted! (We must also remember, most people today don't boil meat in milk. And, truly, what are the chances that the beef and cheese on your cheeseburger came from the same cow?) Take a look at this passage that shows that meat and dairy were indeed served together in "Biblical times."
Genesis 18:1 1 ADONAI appeared to Avraham by the oaks of Mamre as he sat at the entrance to the tent during the heat of the day. 2 He raised his eyes and looked, and there in front of him stood three men. On seeing them, he ran from the tent door to meet them, prostrated himself on the ground, 3 and said, "My lord, if I have found favor in your sight, please don't leave your servant. 4 Please let me send for some water, so that you can wash your feet; then rest under the tree, 5 and I will bring a piece of bread. Now that you have come to your servant, refresh yourselves before going on.""Very well," they replied, "do what you have said." 6 Avraham hurried into the tent to Sarah and said, "Quickly, three measures of the best flour! Knead it and make cakes." 7 Avraham ran to the herd, took a good, tender calf and gave it to the servant, who hurried to prepare it. 8 Then he took curds, milk and the calf which he had prepared, and set it all before the men; and he stood by them under the tree as they ate. (CJB)
All that being said, we don't go about to cause a brother to stumble. So for those who wish to adhere to the rabbinical additions to the commandment, extending it to "do not mix meat and dairy", we should not be stuffing a cheeseburger in our mouths when we are in their company.