Fasting is a cleansing process. By relinquishing the needs of the body and our animalistic nature, we cleanse ourselves of undesirable thoughts and deeds. Engaging in lofty matters enables us to repent completely.

Cleansing the soul of all confusion:

The solemn fast day: On this day, one becomes purified from every sin he has committed in the past and completes anything not achieved on Shabbatot and Rosh Hodesh [which are also days of purification]. On this day, the individual is purified of all confusions of the mind, the forces of anger and desire, and will wholly repent from holding on to them in thought or deed. If it is impossible for him to remove them from his mind, because his habitual thinking that he has retained since his youth overcomes him, as a result of songs he has heard or from stories and the like, then it is sufficient that he become purified in his actions, and that he seek to atone for his thoughts and accept upon himself not to bring them to his lips, as it is said: “I pondered, and nothing passed my lips” (Psalms 17:3), and all the more so, not to express them through any action.

What is the benefit of fasting?

The fast on this day makes the pious one almost like the angels, as the day passes by in humility and in submission, while standing and bowing, praising and giving thanks. All of one’s physical powers abstain from the natural needs [of the body] and are occupied only with fulfilling mitzvot, as if he did not have an animalistic nature at all. (Rav Yehuda HaLevi, Kuzari 3:5)

On one day a year, we come closer to God and to holiness. In order to receive the special light of this day, all fast on Yom Kippur, emulating the angels to some degree.

The purpose of Yom Kippur:

The idea of Yom Kippur is that the Master, blessed be He, prepared one day for Israel when repentance is readily accepted and sins are easily wiped away. That is to say, it is to repair all the damage that has been done and to remove all the darkness that has taken over as a result, and to bring those who repent back to the level of holiness and closeness to God, from which they were distanced as a result of their sins.

The way to receive the special light of Yom Kippur is through fasting:

On this day there is a light shining, by whose power this entire matter [of coming close to God, and of the atonement for sins] is achieved. In order to receive this light, Israel must observe that which they were commanded concerning this day, and in particular with regard to afflicting [the body], through which they become thoroughly disconnected from physicality, and ascend a little toward resembling the angels. (Rabbi Moshe Ĥayyim Luzzatto, Derekh Hashem 4:8:5)