All that we have has come from others, especially God. The Day of Atonement points out how needy and dependent on God we are; fasting shows our frailty.
Atonement, when we are commanded to afflict our souls, is a time of self-evaluation and repentance. This is the only way to have real unity with God.
The intent of fasting is to deflate our pride—the major taproot of sin—the biggest deterrent to a positive relationship with God. Humility heals the breach.