Peter Leithart has a series of posts on the power of blood in atonement rituals.
In The Gift of Efficacy, Leithart writes:
Blood atones. That’s rooted in the created fact that life is in the blood, but in the Bible this doesn’t mean that blood is a kind of magical moral detergent. How then can it atone? It atones because God has “given” it for atonement: “I have given [blood] to you on the altar to make atonement for your lives” (Leviticus 17:11). . . . And so we extrapolate: Baptismal water is given for remission of sins because God gave it for the remission of sins. Bread is an appropriate food, but it is Eucharistic bread is [sic] the bread of heaven because Jesus us gave it to us as such, because He said “This is my body.” Wine is the blood of the grape, but Eucharistic wine is the blood of Christ because Jesus said “This is the covenant in My blood.” No transubstantiations here. No electrification of created things with supernatural juice. It’s enough that they are gifts gifted to be effective.
Why did blood atone? Because God says it did. God “gave” it for atonement.