Why does the Day of Atonement point to Jesus Christ? How did He fulfill this day by His work on the cross?
The Day of Atonement
This was the one day whereby all the nation of Israel was commanded to fast and was the only feast day of the Jews where there was no food to be eaten from the beginning of this High Day at sunset to sunset the following day. It was a solemn feast day for them and one that symbolized the taking away of sin and placing the guilt on a goat and sending him off into the Wilderness. Jesus conquered the very best of Satan’s temptation and He was 40 days and 40 nights in the Wilderness overcoming the strongest tempter of all time. The Orthodox Jews call this the Jewish High Day of Atonement or Yom Kipper (Yom, “day” and Kipper, “to atone”) and today Christians see this fulfilled by Christ Who has made us “at one” with God through His own bodily sacrifice. The Day of Atonement was a shadow of the coming work of Christ on Calvary; therefore this day is no longer needed because there remains no more sacrifice for sins (Heb 8:5).
This feast day was the only one where the entire animal is burned on the altar, except for the animal skin which the priest would keep. This shows the complete and utter sacrifice of Jesus and the fire that consumed the animal shows that God’s judgment was placed upon Christ (as fire is symbolic of God’s judgment) on our behalf. The animal skin was all that remained and the skin could be worn by the priest, but certainly not while on duty. This which reflects that we are clothed in His righteousness and when God the Father sees us, He actually sees Jesus’ own righteousness (2 Cor 5:21). It is as Isaiah wrote “he has clothed me with the garments of salvation; he has covered me with the robe of righteousness” (Isaiah 61:10). This is very much like Zachariah 3:4-5 where the angel of the Lord told Joshua (not Joshua from the Book of Joshua) the high priest “Remove the filthy garments from him.” And to him he said, “Behold, I have taken your iniquity away from you, and I will clothe you with pure vestments.” And I said, “Let them put a clean turban on his head.” So they put a clean turban on his head and clothed him with garments.”
Jesus Offering of Himself
This Day of Atonement was a Burnt Offering and was considered the most important of the Jewish High Days because of the significance of assigning the guilt on the scape goat and placing hands on the goats head and then sending him off into the Wilderness, thus sending away the sins of the people. If the goat came back it was sometimes pushed off a cliff and killed showing that this guilt was permanently removed, just as Christ’s death has once and for all removed our sins from us. Paul said that these days were a shadow of things to come and the One Who cast the shadow is now here in reality and the Light of the World that cast that shadow was Jesus Christ (Col 2:16-17). Certainly the light is more important than the shadow because this Light was the source or cause for the shadow to be cast. That Light of course was Jesus Himself Who is called the light of the world. John wrote “about the light, that all might believe through him. He was not the light, but came to bear witness about the light. The true light, which gives light to everyone, was coming into the world” (John 1:7-9).
The Scape Goat and Jesus’ Becoming Sin
Leviticus shows us the solemn ceremony that was involved in ascribing the sin guilt of the people onto the goats head in chapter 16:21-22 “And Aaron shall lay both his hands on the head of the live goat, and confess over it all the iniquities of the people of Israel, and all their transgressions, all their sins. And he shall put them on the head of the goat and send it away into the wilderness by the hand of a man who is in readiness. The goat shall bear all their iniquities on itself to a remote area, and he shall let the goat go free in the wilderness.”
Here is what seems to match what Christ did on the cross. Aaron, the high priest, laid both hands on the head of the goat, the seat of intellect, and then confessed over it all of their sins. Today we confess our sins and our Intercessor, Jesus Christ the High Priest enters into the Holy of Holies like Aaron did. He sprinkled the blood of a bull over the mercy seat just as Jesus entered the Holy of Holies with His own blood and we can confess our sins as they were placed on Christ and He intercedes for us. We now have access to the Holy of Holies, where the Father resides in heaven and we can approach Him by the blood born sacrifice of Jesus (Heb 9:12). That is why they were told “afflict yourselves” (Lev 16:31) because Jesus was more than afflicted for our redemption. Afflict almost always means fasting. Today, believers fast  when they have sinned grievously, are praying for a great need, or just hunger to know God better.
Jesus the High Priest
Hebrews chapters eight and nine give a beautifully vivid portrait of the eternal High Priest, Jesus Christ’s work.
Hebrews 8:6-7, 13 “it is, Christ has obtained a ministry that is as much more excellent than the old as the covenant he mediates is better, since it is enacted on better promises. For if that first covenant had been faultless, there would have been no occasion to look for a second. In speaking of a new covenant, he makes the first one obsolete. And what is becoming obsolete and growing old is ready to vanish away”
The first covenant was not so as much faultless as it was not able to take away sins; it could only cover them. Jesus’ blood sacrifice made them obsolete and takes sins away.
Hebrews 9:12 “he entered once for all into the holy places, not by means of the blood of goats and calves but by means of his own blood, thus securing an eternal redemption.”
As there used to be daily offerings of blood sacrifices and annual High Days where animals had to be killed, showing the high price of sin, they could not take the sins away because the blood of goats (even the scape goat) and calves (or bulls) had to be repeated over and over again but Jesus entered the holy place one time with His own blood and secured our “eternal redemption.”
Hebrews 9:24-26 “For Christ has entered, not into holy places made with hands, which are copies of the true things, but into heaven itself, now to appear in the presence of God on our behalf. Nor was it to offer himself repeatedly, as the high priest enters the holy places every year with blood not his own, for then he would have had to suffer repeatedly since the foundation of the world. But as it is, he has appeared once for all at the end of the ages to put away sin by the sacrifice of himself.”
The temple, a copy of the heavenly, was not the same as the literal Holy of Holies in heaven so Jesus entered there Himself, bringing in His own blood in the “presence of God on our behalf” and it was done once and for all. Since He was the Perfect Sacrifice, this didn’t have to be done over and over. This “put away [our] sin by the sacrifice of himself.”
Hebrews 9:26 says that “he has appeared once for all at the end of the ages to put away sin by the sacrifice of himself” but then tells us that “it is appointed for man to die once, and after that comes judgment” (Heb 9:27). You have an appointment with death and you have no idea when that will be but “Christ, having been offered once to bear the sins of many, will appear a second time” (Heb 9:28a) and if Christ’s blood sacrifice was not accepted by you then you will be cast into the lake of fire (Rev 20:11-15) but if you have repented and trusted in Him , He will appear a second time but “not to deal with [your] sin but to save [you] who are eagerly waiting for him” (Heb 9:28b). What you have coming to you at His second coming is drastically different. What will it be for you?
Something more to interest you: 7 Good Reasons Christians Should Fast 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), copyright © 2001 by Crossway, a publishing ministry of Good News Publishers. Used by permission. All rights reserved.