The Talmud tells us (Taanis 30b) that every year on Yom Kippur we commemorate the fact that on that very day, so many years ago, Moshe Rabbeinu came down from Mt. Sinai with the second set of luchos. But we also know that every year on Shavuos, both in the davening and in the Kiddush, we identify that holiday as "z'man mattan Toraseinu." Why do we need two holidays for the sake of commemorating the same thing - our receiving of the Torah?
R. Betzalel Hakohein (the dayan of Vilna) suggested that perhaps on Shavuos we commemorate our receiving on the Torah Shebiksav, while on Yom Kippur we commemorate our receiving of the Torah Shebeal Peh (see Nefesh Horav, pg 293.)
Rav Soloveitchik explained that this suggestion is not merely an arbitrary "teratz", that there are two holidays because there are two parts of the Torah, but is really a "milsa debetaama." In the essays of the Beis Halevi a thesis is developed based on many passages in the Talmud, the Midrash, and the Zohar, that on Shavuos, when Hashem proclaimed the aseres hadibros, the plan was to give Moshe Rabbeinu the luchos which would contain all of the Torah on them. According to this original plan, there would be no need for any Torah Shebeal Peh. Everything would appear on the luchos. After the Jewish people sinned with the eigel, they were weakened to the point that now it would be possible for other nations to dominate over them. Those other nations might also possibly dominate over the Torah as well, and claim that they are "the chosen nation", since they have the Torah. For this reason, G-d instituted the Torah Shebeal Peh, which would only remain transmitted among the Jews. In this way, the Jewish people would maintain their uniqueness and their chosen-ness, by virtue of the fact that they alone have this oral transmission of this Torah Shebeal Peh. And this is what the prophet Hoshea (8:12) was referring to: "If I were to have the entire Torah committed to writing, then the enemies of the Jewish people would be able to claim that the Jews were 'strangers', that they had lost their status as 'am hanivchar'; and that that they (the enemies) were now the chosen people!" ( see Gittin 60b).
So the Rov explained, that on Shavuos, when we commemorate maamad Har Sinai, and our receiving of the Torah the first time, this really relates only to our receiving the written Torah; because according to the first plan, there wasn't going to be any oral Torah at all. Following plan #1, there wouldn't have been any need for it. However by the time Moshe Rabbeinu came down on Yom Kippur with the second set of luchos, the entire plan had changed, and that Yom Kippur was the beginning of the Torah Shebeal Peh (see Yemei Zikaron, p. 245.)
In that same prophecy of Hoshea (8:10) the navi encourages the Jewish people that "if they will emphasize the study of Mishna (i.e., Torah Shebeal Peh), G-d will redeem them" (see Midrash Vayikra Rabba 7:3.) The rabbis had a tradition that G-d who instilled within all of us a yetzar horah, also gave us the Torah to serve as an antidote to that yetzer horah (Kiddushin 30b). Until the period of the Anshei Keneses Hagdolah, the dominant yetzer horah was for avodah zarah (see Sanhedrin 102b), and apparently the main antidote for that yetzer horah was the study of Torah Shebiksav. After the Anshei Keneses Hagdolah succeeded in abolishing the yetzer horah for avodah zarah through their tefillos (Yoma 69b), a new yetzer horah was instilled within us for "minus" and " apikursus." The main antidote for that yetzer is to emphasize the study of Torah Shebeal Peh (see B'ikvei Hatzon p. 139.) The "Seder Olam" records that the death of Chagai, Zecharia, and Malachi was the end of the period of the prophets, (i.e. the end of the period of the Torah Shebiksav.) From that point we were instructed to bend our ears and pay good attention to what the rabbis have to tell us (i.e., the Torah Shebeal Peh.) The navi Hoshea was alluding to our period of history when he spoke of emphasizing the study of Mishna and Torah Shebeal Peh.
Regarding Torah Shebeal Peh, the key word is "mesorah." The attitudes and the style of thinking must be transmitted from rebbe to talmid. The opening mishna in Avos tells us that Moshe received the Torah from G-d at Mt. Sinai, and transmitted it (messara) to Joshua; and each succeeding generation transmitted the Torah Shebeal Peh to the next generation. There can not be Torah Shebeal Peh without " mesorah." One lacking such a mesorah can not sit down with a sefer of mishnayos or gemorrah and come up with some new ideas and claim that this is in the spirit of the Torah Shebeal Peh. Strictly speaking, there is no text to Torah Shebeal Peh. It is a system of ideas and attitudes giving an approach which was intended to be transmitted orally - along with the full depth and flavor of meaning and understanding of those attitudes and ideas. At one point in history, the Rabbis were afraid that due to the many persecutions and exiles, much of the oral Torah would be forgotten, so they felt compelled to preserve it by writing it down. But that text can not really stand alone. It requires a strong mesorah to understand what the text (of the Talmud) is driving at. The mesorah did not end when R. Yehuda Hanasi edited the mishna; nor did it end when Ravina and Rav Ashi edited the gemorrah. The mesorah has extended to our generation, and will continue to be transmitted on.
From the very beginning and throughout the entire period of the second temple, there were groups who challenged the mesorah of the Oral Torah. In later years there were Karaites, and yet later - the Haskalah movement. As we say in the Haggadah, "bechol dor vodor, omdim aleinu lechalosienu." The navi Hoshea has warned us that in our period of history, in order to maintain our identity and not get washed away in assimilation, we must emphasize mesorah of the Oral Torah.
Often there are mesorahs which we find difficult to understand, or difficult to swallow. Parts of the Torah Shebeal Peh seem not to be politically correct. Rav Soloveitchik said over a homiletic interpretation of the passage in the gemorrah (Menachos 29b), that Rabbi Akiva, rather than be apologetic, would be more meticulous and place extra emphasis on all of those halachos where the enemies of Torah had thrown thorns. Rather than discard anything that at first glance we are uncomfortable with, we must preserve our mesorah, and try to develop a deeper insight into what it represents. The superficial mind will often misunderstand Torah, and cast away very precious traditions.
This added theme of Yom Kippur as being the day to commemorate the start of the Torah Shebeal Peh was especially obvious during the period of the Second Temple. Every Yom Kippur, the rabbis would make the Kohein Gadol swear that he would not deviate from the oral tradition in doing the avodah.
Many years later, the Orthodox Jewish community of Alexandria would have an annual march - on Yom Kippur - to declare that they subscribed to the Torah Shebeal Peh. Rav Soloveitchik felt that our practice to recite the lengthy seder ho'avodah in chazoras hashatz of mussaf is probably also for the same purpose - to reaffirm our commitment to the mesorah and the Torah Shebeal Peh (see Lustiger, "Before Hashem you Shall be Purifed" p. 144.)