The prohibition from eating the meat of strangled animals is essentially the same thing as the prohibition from eating blood. If an animal was strangled, that means its blood was not drained from its body. Therefore, eating the meat of strangled animals was another method of consuming their blood. The Mosaic Law outlawed the consumption of blood.
Lev. 17:10-11 (NASB)
And any man from the house of Israel, or from the aliens who sojourn among them, who eats any blood, I will set My face against that person who eats blood and will cut him off from among his people. For the life of the flesh is in the blood, and I have given it to you on the altar to make atonement for your souls; for it is the blood by reason of the life that makes atonement.
Deut. 12:16 (NASB)
Only you shall not eat the blood; you are to pour it out on the ground like water.
Deut. 12:23 (NASB)
Only be sure not to eat the blood, for the blood is the life, and you shall not eat the life with the flesh.
Since eating blood was so detestable to the Jews, and presumably associated with pagan rituals, the Apostles felt that it was important for the gentile Christians to abstain from eating blood, even though they were not following the rest of the Mosaic Law. By doing so, they would not cause offense to Jews and weaker brethren.
1 Cor. 10:23-33 (NASB)
All things are lawful, but not all things are profitable. All things are lawful, but not all things edify. Let no one seek his own good, but that of his neighbor. Eat anything that is sold in the meat market without asking questions for conscience’ sake; for the earth is the Lord’s, and all it contains. If one of the unbelievers invites you and you want to go, eat anything that is set before you without asking questions for conscience’ sake. But if anyone says to you, “This is meat sacrificed to idols,” do not eat it, for the sake of the one who informed you, and for conscience’ sake; I mean not your own conscience, but the other man’s; for why is my freedom judged by another’s conscience? If I partake with thankfulness, why am I slandered concerning that for which I give thanks? Whether, then, you eat or drink or whatever you do, do all to the glory of God. Give no offense either to Jews or to Greeks or to the church of God; just as I also please all men in all things, not seeking my own profit but the profit of the many, so that they may be saved.
Some people take this command further and believe that, even today, it is a sin for Christians to eat blood. They believe that, since God outlawed the consumption of blood prior to the Mosaic Law, it is a general moral law which applies to all people for all time, not just those subject to the Mosaic Covenant.
Gen. 9:4-5 (NASB)
Only you shall not eat flesh with its life, that is, its blood. Surely I will require your lifeblood; from every beast I will require it. And from every man, from every man’s brother I will require the life of man.