Eat no fat or blood

Lev 3:17 It shall be a perpetual statute for your generations throughout all your dwellings, that ye eat neither fat nor blood.

The pagans ate fat and drank blood in worship of their false gods.

Deu 32:37 And he shall say, Where are their gods, their rock in whom they trusted,
Deu 32:38 Which did eat the fat of their sacrifices, and drank the wine of their drink offerings? let them rise up and help you, and be your protection.
Deu 32:39 See now that I, even I, am he, and there is no god with me: I kill, and I make alive; I wound, and I heal: neither is there any that can deliver out of my hand.

The disobedient leaders ate the fat in rebellion and did not feed their flock.

Eze 34:2 Son of man, prophesy against the shepherds of Israel, prophesy, and say unto them, Thus saith the Lord GOD unto the shepherds; Woe be to the shepherds of Israel that do feed themselves! should not the shepherds feed the flocks?
Eze 34:3 Ye eat the fat, and ye clothe you with the wool, ye kill them that are fed: but ye feed not the flock.

There was quite a stir when Jesus said that we must drink his blood. The communion wine was symbolic of oneness with Jesus, his life in us.

Joh 6:51 I am the living bread which came down from heaven: if any man eat of this bread, he shall live for ever: and the bread that I will give is my flesh, which I will give for the life of the world.
Joh 6:52 The Jews therefore strove among themselves, saying, How can this man give us his flesh to eat?
Joh 6:53 Then Jesus said unto them, Verily, verily, I say unto you, Except ye eat the flesh of the Son of man, and drink his blood, ye have no life in you.
Joh 6:54 Whoso eateth my flesh, and drinketh my blood, hath eternal life; and I will raise him up at the last day.
Joh 6:55 For my flesh is meat indeed, and my blood is drink indeed.
Joh 6:56 He that eateth my flesh, and drinketh my blood, dwelleth in me, and I in him. Joh 6:57 As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me.
Joh 6:58 This is that bread which came down from heaven: not as your fathers did eat manna, and are dead: he that eateth of this bread shall live for ever. 

The fat of a sacrifice was burned as an offering. It made a sweet smell. When we barbecue meat, the fat burns off and the smoke of it gives flavor to the meat. In one sense the sweet smell is our sweet smelling offering, as incense is an offering of prayer. There is a modern slang "chew the fat", an English expression for gossiping or making friendly small talk. When we come together it is better to partake of the fellowship of Christ in sharing and prayer. The blood of a sacrifice is to be poured out, because the life is in the blood. Many would not eat meat of an animal sacrificed to an idol. Jesus is the living presence of God given as a sacrifice for us. We are to eat the bread of life and drink the symbolic blood in communion, for he is our life through his death and resurrection for us.

1Co 11:24  And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me. 
1Co 11:25  After the same manner also he took the cup, when he had supped, saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me. 
1Co 11:26  For as often as ye eat this bread, and drink this cup, ye do shew the Lord's death till he come. 

 We may eat the symbolic bread (meat) of the body of Christ and drink of His blood, the communion grape juice, because he is life to us and we come together being part of the body of Christ. Taste and see that the Lord is good. (psalm 34:8)