The Previous Rebbe once recalled that at his father’s (the Rebbe Rashab’s) minyan on the night of Yom Kippur one facet of the service was simcha (joy).
Our Rebbe comments that, at first glance, it is surprising to hear that Yom Kippur night was a time of simcha because Yom Kippur is known as a time of tshuva.
The Rebbe resolves the apparent contradiction by explaining that the tshuva of Yom Kippur reaches a very high level of tshuva [tshuva ilah].
The Rebbe points out that the Rambam alludes to this in Hilchos Teshuva, Halachos Six and Seven.
Halacha Six states “Even though teshuva and calling out (to G-d) are always desirable, they are even more desirable during the ten days between Rosh Hashana and Yom Kippur, when the teshuva and calling out will be accepted immediately, as Isaiah 55:6 states, “Seek G-d when he is to be found. To whom does this apply? To an individual. However, regarding a community, whenever they repent and cry out sincerely, they are answered immediately, as Devorim 4:7 states, “What nation is so great that they have G-d close to them, as G-d, our L-rd, whenever, we call to him.”
Halacha Seven states, “Yom Kippur is the time of general Teshuva for both individuals and the community. It is the ketz mechila u’slicha l’Israel, the endpoint and conclusion of forgiveness and pardon for Israel.”
Since the service of Yom Kippur reflects the level of “tshuva ilah,” that is why it must be expressed by simcha. This can be seen by the well-known statement of Koheles 9:7, “At the end of Yom Kippur, a heavenly voice announces, ‘go and eat your bread in happiness [simcha] because Hashem has accepted your actions with good will.’