The interpretation of the sacrificial death of Jesus by the Roman Catholic Church and many others is the one proposed by St. Anselm, namely that Jesus paid by his death for the sins of humanity. It echoes the passage in the letter to the Hebrews (9:11-14) which sees him as taking place of the "goats and calves" sacrificed by a priest on the Day of Atonement.
But I do not believe that that sacrificial ritual relates to the death of Jesus. He did not die on the Day of Atonement but on the Day of Passover and identified not with the "goats and calves" but with the Passover Lamb. The sacrifice of the Lamb was a means of escaping from captivity in Egypt, just as we may escape from sin and death by sacrificing our selfish acts and desires. Jesus's sacrifice is not a payment for sins but an example pointing a way out of the captivity of sinful selfishness. It is a culmination and validation of the entire body of his teaching.
The theory of "payment" is based on the view that evil in the world is the result of human wickedness, of a debt incurred by Adam and his descendants. But in reality evil is needed for the definition of good, and death for the realization of life. God exposes the world, his beloved child, to suffering and death so that it may live. This is symbolized by the suffering of Jesus, the Son of God, and by his Resurrection.
Day of Atonement
This day was in a sense, the most sacred of the year for Israel as it was the day when the High Priest entered the Most Holy Place; it happened once a year (Leviticus 16 & Numbers 29:7-11).
He entered first to make atonement for his own sin, by blood sacrifice and then another blood sacrifice for the sins of the nation of Israel. ‘Without the shedding of blood there is no forgiveness of sin’ Hebrews 9:22 & (Leviticus 17:11).
This feast was actually a day of afflicting one's soul, humiliation and a day of no work (Leviticus 16:29-31); all of which are fulfilled in Christ, who has forgiven us when we confess our sins and is our Sabbath Rest.
Jesus is our ‘Great High Priest’, ‘our mediator’, and He is also the spotless ‘Lamb of God who takes away the sin of the world’ (Hebrew 9:11 & 1 Timothy 2:5 & John 1:29).
Jesus, as the High Priesthood of Melchizedek forever; once and for all, made atonement for mankind, unlike the priest from Aaron’s lineage who had to come once a year, every year to sacrifice by the blood of animals, and lay sin on the goat outside the gate. The purpose of this feast was to cleanse the people from their sin and to purify the Holy Place. Jesus is the most holy place [The Temple], and complete atonement for sin, satisfying and fulfilling the shadow of the Law. [Colossians 2:16-18; Hebrews 8:5; Hebrews 10:1]
Romans 3:24-26, speaks of Christ’s redemption by His blood and Hebrews 9:7, 10:3, 19-22 speaks of Jesus as the mediator and that we can enter the Holy of Holies by the blood of Jesus.
The following corresponding verses from the Mosaic Law compared with the NT, show exactly how this feast was fulfilled in Christ, fully and completely.
[Leviticus 16 corresponds with Leviticus 23:27-32]
Lev 16:3 Thus shall Aaron [the High Priest] come into the holy place: with a young bullock for a sin offering, and a ram for a burnt offering.
Heb 9:11,12 But Christ being come an high priest... Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us.
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Lev 16:12,13 And he shall take a censer full of burning coals...and bring it within the veil: And he shall put the incense upon the fire before the LORD, that the cloud of the incense may cover the mercy seat...
Heb 6:19,20 ...which entereth into that within the veil; Whither the forerunner is for us entered, even Jesus, made an high priest for ever after the order of Melchisedec.
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Lev 16:15,16 Then shall he kill the goat of the sin offering, that is for the people, and bring his blood within the veil, and do with that blood as he did with the blood of the bullock, and sprinkle it upon the mercy seat, and before the mercy seat: And he shall make an atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions in all their sins...
Heb 9: 13,14,23,24 For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifieth to the purifying of the flesh: How much more shall the blood of Christ, who through the eternal Spirit offered himself without spot to God, purge your conscience from dead works to serve the living God? ... It was therefore necessary that the patterns of things in the heavens should be purified with these... For Christ is not entered into the holy places made with hands, which are the figures of the true; but into heaven itself, now to appear in the presence of God for us:
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Lev 16:21,22 And Aaron shall lay both his hands upon the head of the live goat, and confess over him all the iniquities of the children of Israel, and all their transgressions in all their sins, putting them upon the head of the goat, and shall send him away by the hand of a fit man into the wilderness: And the goat shall bear upon him all their iniquities unto a land not inhabited: and he shall let go the goat in the wilderness.
Isa 53:6,11 ...the LORD hath laid on him the iniquity of us all...for he shall bear their iniquities.
Heb 9:28; 13:12,13 So Christ was once offered to bear the sins of many... Wherefore Jesus also, that he might sanctify the people with his own blood, suffered without the gate. Let us go forth therefore unto him without the camp, bearing his reproach.
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Lev 16:30 For on that day shall the priest make an atonement for you, to cleanse you, that ye may be clean from all your sins before the LORD.
Heb 10:10,12,14 By the which will we are sanctified through the offering of the body of Jesus Christ once for all. But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God; For by one offering he hath perfected for ever them that are sanctified.
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Luke 24:44-48
And he said unto them, These are the words which I spake unto you, while I was yet with you, that all things must be fulfilled, which were written in the law of Moses, and in the prophets, and in the psalms, concerning me. Then opened he their understanding, that they might understand the scriptures And said unto them, Thus it is written, and thus it behooved Christ to suffer, and to rise from the dead the third day: And that repentance and remission of sins should be preached in his name among all nations, beginning at Jerusalem. And ye are witnesses of these things.
