Why did God require blood sacrifices in the Old Testament?

People cannot be saved by the blood of animals. The animal sacrifices in the Old Testament were merely a foreshadowing of the sacrificial death of God the Son. God sees everything in the present tense (He is beyond space and time as we know it). The blood of Christ applies to anyone who accepts Christ today, and applied to everyone who had faith in God and His promised Messiah, all the way back to Adam. The sacrifice of animals was only a way of showing (i.e., foreshadowing) the death of Christ before it actually happened. God is not appeased, nor does He enjoy, the shed blood of animals. But then, neither does He enjoy sin. But the Son was willing to die in order to save a remnant of His human creatures from sin, and the fact that the Father allowed the Son to be crushed at the cross shows us how seriously God takes the offense of sin.

As already stated, the Old Testament sacrifices were a foreshadowing---a sort of temporary replacement---to suffice until the real, true, perfect and final sacrifice was enacted, fulfilling God's promise. The blood of animals was only a representation of the real thing (Christ) until the real thing came.

"The law is only a shadow of the good things that are coming—not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship. If it could, would they not have stopped being offered? For the worshipers would have been cleansed once for all, and would no longer have felt guilty for their sins. But those sacrifices are an annual reminder of sins, because it is impossible for the blood of bulls and goats to take away sins. Therefore, when Christ came into the world, he said: "Sacrifice and offering you did not desire, but a body you prepared for me; with burnt offerings and sin offerings you were not pleased. Then I said, 'Here I am—it is written about me in the scroll—I have come to do your will, O God.' " First he said, "Sacrifices and offerings, burnt offerings and sin offerings you did not desire, nor were you pleased with them" (although the law required them to be made). Then he said, "Here I am, I have come to do your will." He sets aside the first to establish the second. And by that will, we have been made holy through the sacrifice of the body of Jesus Christ once for all." (Hebrews 10:1-10)

(Also see "Why was eating blood forbidden? And why blood sacrifices?")