God required animal sacrifices in the Old Testament because there needed to be a way to demonstrate the severity of sin, the cost of rebellion against God, the death that sin brings, and the cost and must be paid to be redeemed. They served as a kind of placeholder while people waited for the true Messiah to come and the true sacrifice would be offered. After all, the blood of animals cannot cleanse from sin (Hebrews 10:4). So, the Old Testament animal sacrifices were a representation of the death of Christ on the cross that would later occur and which was recorded in the Gospels.
Hebrews 10:1-3, "For the Law, since it has only a shadow of the good things to come and not the very form of things, can never, by the same sacrifices which they offer continually year by year, make perfect those who draw near. 2 Otherwise, would they not have ceased to be offered, because the worshipers, having once been cleansed, would no longer have had consciousness of sins? 3 But in those sacrifices there is a reminder of sins year by year.
In the Old Testament the bloodshed was necessary because God told us in Leviticus 17:11, "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." Atonement is a cleansing, a removal of sin. So, animal sacrifices were a temporary means by which atonements could occur that would "cleanse" people of their sins. They were fulfilled in Christ and with him, no further animal sacrifices were necessary.
Hebrews 9:11-14, "But when Christ appeared as a high priest of the good things to come, He entered through the greater and more perfect tabernacle, not made with hands, that is to say, not of this creation; 12 and not through the blood of goats and calves, but through His own blood, He entered the holy place once for all, having obtained eternal redemption. 13 For if the blood of goats and bulls and the ashes of a heifer sprinkling those who have been defiled, sanctify for the cleansing of the flesh, 14 how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?
To be in God's holy presence, we must be without sin because God is Holy (1 Peter 1:16). When a person sins there is a consequence for breaking the Law of God. That consequence is death (Genesis 2:17; Exodus 20:1-17; Romans 6:23) and separation from God (Isaiah 59:2). On the cross, Jesus bore our sins in his body (1 Peter 2:24). Another way of looking at it is that he "became sin" (2 Corinthians 5:17), though he never committed sin (1 Peter 2:22). So, when he died on the cross, the Law of God was satisfied; this is justice. But, because God is merciful and gracious he extends that death, that atoning sacrifice to all who would receive it by faith, so that all believers are made right before God (Romans 5:1; Philippians 3:9). With Christ's death, a new covenant was instituted.
Heb. 9:15, "And for this reason He is the mediator of a new covenant, in order that since a death has taken place for the redemption of the transgressions that were committed under the first covenant, those who have been called may receive the promise of the eternal inheritance.
Therefore, because Jesus died on the cross, we had a new covenant, a new contract with God where God gives us the righteousness of Christ (Philippians 3:9) and deliverance from eternal judgment.