Like many of God's holy days, Atonement is rich in meaning, and it has more than one meaning. It represents the putting away of Satan so he can no longer deceive the nations (Leviticus 16:10, 20-26, 29-31, Revelation 20:1-3). It also, like Passover, represents the sacrifice of Jesus Christ for our sins (Leviticus 16:7-9, 15-19).
There is an important lesson in the Day of Atonement about overcoming Satan and his influences. There is a connection made between fasting and overcoming Satan.
The Day of Atonement, the day that represents the putting away of Satan, is the only day God commands us to fast on. Fasting is an effective tool for overcoming the temptations and wrong influences that Satan uses to attack us and try to destroy us spiritually.
When Jesus went into the wilderness to face the greatest temptations that Satan could throw at Him, He fasted for forty days (Matthew 4:1-2, Luke 4:1-2). Though He was physically weakened by that ordeal, He was close to His Father in heaven. He was able to resist all of Satan's temptations.
When Jesus's disciples tried to cast out a particularly resistant demon and could not, Jesus said that this kind of demon can only be cast out through prayer and fasting (Matthew 17:14-21, Mark 9:17-29).
The lesson in all this seems to be that prayer and fasting are valuable tools for overcoming Satan, temptation, and sin. For as long as I have been in the Church, and I am sure before that, the Church of God has taught fasting as a valuable tool for drawing closer to God, and the Bible confirms this. Perhaps God has given us Atonement so we can experience fasting as an encouragement to fast regularly throughout the year as a tool for drawing closer to God and overcoming sin.
Here are links to related sections in Preaching the Gospel:
The Day of Atonement - the Putting Away of Satan, Chapter 2