The Day of Atonement involves not only the forgiveness of sin; it pictures the removal of the primary cause of sin, Satan. It teaches us that Jesus Christ gave His life to atone for the sins of all mankind. It also points to the time when Satan will be bound for 1,000 years (Leviticus 16:29-30 Leviticus 16:29-30 [29] And this shall be a statute for ever to you: that in the seventh month, on the tenth day of the month, you shall afflict your souls, and do no work at all, whether it be one of your own country, or a stranger that sojournes among you: [30] For on that day shall the priest make an atonement for you, to cleanse you, that you may be clean from all your sins before the LORD.
American King James Version×, Leviticus 16:20-22 Leviticus 16:20-22 [20] And when he has made an end of reconciling the holy place, and the tabernacle of the congregation, and the altar, he shall bring the live goat: [21] And Aaron shall lay both his hands on 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 on the head of the goat, and shall send him away by the hand of a fit man into the wilderness: [22] And the goat shall bear on him all their iniquities to a land not inhabited: and he shall let go the goat in the wilderness.
American King James Version×; Revelation 20:1-3 Revelation 20:1-3 [1] And I saw an angel come down from heaven, having the key of the bottomless pit and a great chain in his hand. [2] And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years, [3] And cast him into the bottomless pit, and shut him up, and set a seal on him, that he should deceive the nations no more, till the thousand years should be fulfilled: and after that he must be loosed a little season.
American King James Version×). This Holy Day pictures our High Priest, Jesus Christ, making atonement for our sins, which allows us to be reconciled to God and enter into the "holiest of all" (Hebrews 9:8-14 Hebrews 9:8-14 [8] The Holy Ghost this signifying, that the way into the holiest of all was not yet made manifest, while as the first tabernacle was yet standing: [9] Which was a figure for the time then present, in which were offered both gifts and sacrifices, that could not make him that did the service perfect, as pertaining to the conscience; [10] Which stood only in meats and drinks, and divers washings, and carnal ordinances, imposed on them until the time of reformation. [11] But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building; [12] 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. [13] For if the blood of bulls and of goats, and the ashes of an heifer sprinkling the unclean, sanctifies to the purifying of the flesh: [14] 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?
American King James Version×; Hebrews 10:19-20 Hebrews 10:19-20 [19] Having therefore, brothers, boldness to enter into the holiest by the blood of Jesus, [20] By a new and living way, which he has consecrated for us, through the veil, that is to say, his flesh;
American King James Version×). By fasting on this day, we draw closer to God and picture the reconciliation of mankind to God. Christ is essential in this process as our High Priest (Hebrews 4:14-15 Hebrews 4:14-15 [14] Seeing then that we have a great high priest, that is passed into the heavens, Jesus the Son of God, let us hold fast our profession. [15] For we have not an high priest which cannot be touched with the feeling of our infirmities; but was in all points tempted like as we are, yet without sin.
American King James Version×; Hebrews 5:4-5 Hebrews 5:4-5 [4] And no man takes this honor to himself, but he that is called of God, as was Aaron. [5] So also Christ glorified not himself to be made an high priest; but he that said to him, You are my Son, to day have I begotten you.
American King James Version×, Hebrews 5:10 Hebrews 5:10Called of God an high priest after the order of Melchisedec.
American King James Version×) and as our eternal sacrifice for sin (Hebrews 9:26-28 Hebrews 9:26-28 [26] For then must he often have suffered since the foundation of the world: but now once in the end of the world has he appeared to put away sin by the sacrifice of himself. [27] And as it is appointed to men once to die, but after this the judgment: [28] So Christ was once offered to bear the sins of many; and to them that look for him shall he appear the second time without sin to salvation.
American King James Version×).

To read more about the Day of Atonement click the links below. 

https://www.ucg.org/tags/atonement

Atonement: Removal of Sin's Cause and Reconciliation to God

The Binding of Satan Pictured on the Day of Atonement