It is a big question and, as usual in Judaism, there are diverse interpretations. With regard to shatnez, the best interpretation I’ve read about is based on the book of Genesis and the story of Cain and Abel, in which Cain brought G-d a gift of linen (flax), while Abel brought G-d a gift of wool in the form of sheep. Cain was upset by Abel’s gift and later murdered him. In the Midrash, which is part of the Oral Law (Talmud) necessary to interpret the Written Law (Torah), it is said that this incident shows the importance of people’s respecting each other’s diversity, which Cain was unable to do regarding his brother. By mixing these fabrics, it shows a lack of respect for each person’s individuality and uniqueness. There are other interpretations as well, including the idea that material fabrics represent the spiritual fabric of the Universe and if the former are mixed, they may cause harm to the latter. Finally, there is the idea in Judaism that certain laws were INTENTIONALLY given by G-d without any apparent reason, for the purpose of distinguishing them from those laws that make good sense. This distinction is thought necessary because if every law is commonly shared by humanity and written on the hearts of mankind, one may follow it without necessarily believing in G-d, simply by virtue of being a rational humanist and a good person. However, laws which are given for no apparent reason differentiate those people who believe and trust in G-d and obey His laws despite not understanding their meaning. To the rationalist, this might appear to be nothing but blind obedience without the use of intelligent thinking–which, in general, is not encouraged in Judaism–but in this case, for the believer, it is a sign of one’s love and faith in G-d.
With regard to the prohibition against the mixing of meat and milk, this commandment appears three times in the Torah (twice in Exodus and once in Deuteronomy). As in the case of shatnez, there may be various reasons given by the Oral Law; unfortunately, I’m not familiar with them. I do think, however, that the cultural differences concerning the time period one must wait after eating meat products and then those made with milk CANNOT be divinely inspired. Nonetheless, Orthodox Jews believe in the divine inspiration of BOTH the Written Law and the Oral Law, while Conservative Jews are not as strict about the Oral Law interpretations (that is, they offer differing interpretations), and many Reform Jews go so far as to doubt the divine inspiration of either, preferring to view Judaism as more of an evolving religion than one which is basically a done deal. In principle, this latter view (as well as the one held by Conservative Judaism) are labeled heresies from the perspective of Orthodox Judaism. That does not mean, however, that Torah Jews do not accept and love their heretical brothers as fellow Jews; some of the former believe these uninformed souls will ultimately be “converted” either in the present life or the afterlife.
