Many of the dietary and health regulations listed in Deuteronomy and Leviticus are just plain common sense. It seems that many of the animals, fish, and birds listed as unclean are those who either eat meat or are scavengers.

Leviticus 17:11
For the life of a creature is in the blood, and I have given it to you to make atonement for yourselves on the altar; it is the blood that makes atonement for one's life.
Couldn’t help but think of Jesus on this one. It is through His shed blood that we have atonement for our sins.

Colossians 1:19-20
For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.

While we may not want to commit to memory the qualifications for clean and unclean animals, it is good to have it there for us to refer back to, especially when telling the story of Noah's ark.  Not all the animals went into the ark in pairs of two only.
Genesis 7:2
You shall take with you every clean animal by sevens, the male and female. And take two of the animals that are not clean, the male and female. 

Rather a strange command concerning cooking a young goat in its mother's milk.  Mixtures of milk (or cheese) and meat are prohibited according to Jewish law.  Today, strict Jews do not eat cheeseburgers because it is a mixture of cheese and meat. Jesus freed us from that restriction.
According to the Talmud, these almost identical references are the basis for three distinct dietary laws:

• the prohibition against cooking a mixture of milk and meat
• the prohibition against eating a cooked mixture of milk and meat
• the prohibition against deriving any benefit from a cooked mixture of milk and meat.

And there is also this explanation.
"It was a custom with the ancient heathens, when they had gathered in all their fruits, to take a kid, and boil it in the dam's milk; and then in a magical way, to go about and sprinkle all their trees, and fields, and gardens, and orchards with it, thinking by these means, that they should make them fruitful, and bring forth more abundantly in the following year. Wherefore, God forbad his people, at the time of their in-gathering, to use any such superstitious or idolatrous rite.
Dr. Cudworth, from a MS. comment of a Karaïte Jew.

Many of the strict regulations concerning diet are to demonstrate to the nations that Gods people are to be different, that they are to stand out.
Leviticus 20:26
You are to be holy to Me, because I, the Lord, am holy, and have set you apart from the nations, to be my own.