This is kind of a challenging “answer while standing on one foot” kind of question, but it is worth discussing, because there are many different aspects to Yom Kippur and sources for its observance in the Torah and in stories and tradition. First, we are commanded by the Torah to “afflict ourselves” on Yom Kippur. This is interpreted as fasting, and Yom Kippur is one of the longest fasts in the year (Tisha B’Av is slightly longer) lasting from sunset on the day before to after sundown the night following Yom Kippur (over 25 hours). It is a time to confess our sins, and, more importantly, to renew our connection with G-d. This is why the day is called “Yom Kippur” (or Day of Atonement). It is interesting that in English, Atonement can be broken down into the words At one ment,, or becoming “at one,” united with G-d.
Yom Kippur was also the day that the High Priest would enter the Holy of Holies in the Temple. No one was permitted to enter this place except on this day, and it was only the High Priest who was allowed this privilege. In the Holy of Holies, or the inner chamber of the Temple, there radiated a profound energy that sustained the world, according to Jewish Mysticism, and that is why it was necessary to have the High Priest enter every year. Many high priests did not survive the experience and expired on the spot because the holy energy was so intense. However, the High Priest would enter (and in many generations, replacements would be required) every year.
According to tradition, Yom Kippur is the day that Moses descended from Mount Sinai with a second set of tablets after he had destroyed the first set in the wake of the sin of the Golden Calf. It is said that these tablets radiated an even higher level of holiness than the first set, because they resulted from repentance. This demonstrates the power of atonement and the importance of personal improvement.