In Genesis 3, God Himself slaughters an animal to redeem and cover Adam and Eve because of the first sin.
"18Therefore even the first covenant was not inaugurated without blood. 19For when every commandment had been spoken by Moses to all the people according to the Law, he took the blood of the calves and the goats, with water and scarlet wool and hyssop, and sprinkled both the book itself and all the people, 20saying, “This is the blood of the covenant which God commanded you.” 21And in the same way he sprinkled both the tabernacle and all the vessels of the ministry with the blood. 22And almost all things are cleansed with blood, according to the Law, and without the shedding of blood there is no forgiveness." (Hebrews 9:18-22)
"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.’" (Leviticus 17:11)
It basically comes down to the fact that God is holy, so the penalty for sin (disobedience/evil) is death. Something has to die. God allowed the sacrifice of an animal to pay the penalty. Once Jesus came, lived a perfect life, took our sins on Himself, died to pay the penalty, and rose from the dead to show He had conquered sin and the grave, those who trust in Him no longer need to pay a physical sacrifice because He became our kinsman redeemer and paid the price so we don't have to die.