The mishna at the end of Maseches Taanis records that one of the two jolliest days of the year was Yom Kippur. The Gemorah there gives two reasons for this special joyous atmosphere. One of the reasons is that on Yom Kippur we celebrate the anniversary of our receiving the second set of Luchos. Every Shavuos we celebrate the anniversary of Ma'amid Har Sinai and refer to that day as Zman Matan To'raseinu. Why do we need two yomim tovim celebrating our receiving of the Torah?
The Beis Halevi explains that according to the original plan there was not going to be any Torah Sheb'al Peh, rather everything would have been included in Torah sheb'ksav. Only after the chet ha'eigel, when Hakadosh Boruch Hu decided to punish Bnei Yisroel and allow other nations to rule over them was it necessary to give us a Torah Sheb'al Peh. The umos ha'olam dominating over the Jewish people would be able to claim that we lost our status as am hanivchar and that they took over that special role. The Midrash comments on a posuk (Hoshea 8:12) that if the entire Torah would have been written down the Jewish people would become like "outsiders" and "strangers". Therefore Hashem gave us the Torah Sheb'al Peh which was not supposed to be shared at all with other nations. Through the transmission of the Torah Sheb'al Peh, from generation to generation, we preserved (even in the eyes of the world) our unique status as an am ha'nivchar. We alone had this vast section of Torah which was never shared with anyone else. On Shavuos we celebrate the receiving the Torah sheb'ksav while on Yom Kippur we really celebrate the receiving of the Torah Sheb'al Peh.
In sifrei chasidus an idea is developed, based on a passage in the Zohar, that the Torah sheb'ksav was generally given b'middas hadin while the Torah Sheb'al Peh represent middas hachessed. An obvious example of this would be the way the Chumash describes the punishment for one who maims another person. The simple reading of the text of the Torah sheb'ksav would leave one to believe that we actually maim the assailant, as the possuk reads, "ayin tachas ayin." The Torah Sheb'al Peh teaches us that we should not take that literally; perhaps that is the punishment that the person deserves b'midas hadin, but the middas ha'chessed dictates that instead of maiming him we have him make a cash payment.
The other reason the Gemorah gave why Yom Kippur was a day of such unusually joyous celebration is that Yom Kippur is the day on which Hashem forgives all of our sins. The forgiving of the sins is certainly b'middas hachessed. When we refer to Yom Kippur in the piyutim as a yom hadin ("l'keil orech din, b'yom din") we don't mean din in the sense of strict judgment but rather describe the day as a yom hadin in the sense of a day of calculations (like din v'cheshbon). Yom Kippur is a yom hachessed and not a yom hadin. It certainly fits in that the Torah Sheb'al Peh was given on Yom Kippur since it is the day of chessed and the Torah Sheb'al Peh represents middas hachessed.
Rav Soloveitchik pointed out that according the simple reading of the pessukim in parshas Acharei Mos, the korbonos on Yom Kippur were brought in three units: first there was avodah done by the kohain gadol wearing his golden uniform, then avodah done by kohain gadol wearing his special white uniform, and finally, the third unit of avodas hakorbonos was performed by the kohain gadol wearing the golden uniform. The Torah Sheb'al Peh teaches us that we should not follow the simple reading of the pessukim, and the avodah must be divided into five units and not three. Yom Kippur is the day on which we celebrate our receiving the Torah Sheb'al Peh so perhaps this is the reason why the text of the chumash is so vague on this point and we are required to follow the tradition of the Torah Sheb'al Peh to know the correct way of doing the avodah.