People fast on Yom Kippur because it makes them more dedicated to spiritual exercise. Actually, fasting on something makes you be fully aware of the need to be serious and committed to the very thing you are doing. Yom Kippur means the day of atonement. This day means a lot to Jewish people, and it is considered to be the most important holiday in their calendar.
Yom Kippur is all about seeking God's face. Fasting is recommended for people to show their state of heart about their past mistakes. It is also believed that fasting will make their spirit, soul, and body to be made new before God. And that when God is judging them on Yom Kippur, He will be very merciful with them. There are other spiritual exercises people are expected to commit themselves to. Example of which is giving charity to the less privileged and the needy.