Then G-d instructed Moses, saying, "But on the tenth of this seventh month - it is the day of pardon (yom kippurim) - you shall have a solemn assembly, and you are to torment your souls; and you are to bring a fire-offering to G-d... For any person who is not subjected to torment (humbling) on this very day will be severed from his people." '
(Leviticus 23:26-27, 29)

From this, the Sages derived two Mitzvot relating to Yom Kippur:

(a) Tormenting the soul - a day of fasting;
(b) A ban on acts of melachah.

[Note that kippur, or kippurim , does not mean 'atonement', which is included as a part of Teshuvah (repentance), but rather: pardon, forgiveness, absolution.]

The purpose of the fast is explained in Sefer Hahinuch (Commandment 213) as follows:

'One of the roots of the Mitzvah was that G-d's merciful kindness towards His creatures includes His designation for them with one fixed day in the year on which they may obtain pardon for their sins, through repentance. For this reason, we are commanded to torment ourselves (fast) on that day: for food and drink and other bodily pleasures arouse desire and the temptation to sin, and they discourage the soul from seeking the truth… Nor is it appropriate to arrive and stand in front of the Judge after eating and drinking.'

Sefer Hahinukh also explains the ban on acts of melachah (Commandment 217):

'So that we should not be troubled by any distractions but can direct and concentrate all our thoughts to ask pardon and forgiveness from the Lord.'

Maimonides gives a similar explanation in his Laws of Repentance (4:7):

'Yom Kippur is a time of repentance for both individuals and the community, and is the time of forgiveness and pardon for Israel, so that on Yom Kippur everyone is obliged to repent and to confess (his or her wrongdoings).'

Isaiah (57:14 - 58:14, read on Yom Kippur as the Haftarah) exhorte the People of Israel to amend their ways in life, both in outlook and in deed.

A fast day alone is insufficient, unless accompanied by charging one's ways and deeds (Jonah 3:10).

Moreover, Yom Kippur does not grant forgiveness for wrongdoing against another person, until that other person (the injured party) first grants forgiveness.