Yom Kippur is a day of Matan Torah (Taanis 30); it is the day on which Moshe Rabeinu brought down the second Luchos.  The Beis Halevi (Drasha 18) explains that Yom Kippur is more precisely the day of Matan Torah of Torah Sheba'al Peh. The Beis Halevi maintains that on the first Luchos, Hashem had written down the entire body of Torah Sheba'al Peh. Torah Sheba'al Peh as we know it wouldn't have existed with the first Luchos; it was all written down. The only part of Torah Sheba'al Peh that would have been necessary was the "lehavin davar mitoch davar", to compare a new case to that which is already written in the Torah. This is the Beis Halevi's opinion. 

The Beis Halevi uses this idea to explain many Pesukim and Midrashim.  He says this is the pshat of the Gemara (Eruvin 54) which says regarding the first Luchos that there would have been no "shichecha", no forgetting of the Torah. Why? Because it was all written down and therefore impossible to forget.

However, by the second Luchos, Hashem changed things. By the second Luchos the Torah Sheba'al Peh wasn’t written down on the Luchos; rather, it was literally Ba'al Peh. Only the Torah Shebichtav was written on the Luchos themselves. So, according to the Beis Halevi, the concept of the split of Torah Shebichtav and Torah Sheba'al Peh came into existence for us on Yom Kippur. 

Rabbi Soloveitchik zt"l, the Rav, thinks that one can explain an idea in Parshas Achrei Mos based on this yesod of the Beis Halevi, his great-grandfather (Nefesh Harav p.293). The Torah describes the special Avodas Yom Kippur of the Kohen Gadol at great length with many details. If one reads the Chumash according to the simple pshat, then the Pesukim indicate that the Kohen Gadol went to the Mikvah three times during the Avodah. However, Chazal have a mesorah that the Kohen Gadol went to the Mikvah five times[i]. So this parsha is one of the many examples where the Torah Sheba'al Peh presents a different picture than the simple Torah Shebichtav. The Rav points out, however, that what's happening here is actually uncommon. Generally what happens is that the Torah Shebichtav is not fully clear, and the Torah Sheba'al Peh clarifies what the Torah Shebichtav means. Or, the Torah Shebichtav is brief, and the Torah Sheba'al Peh elaborates on the text. But here in Parshas Achrei Mos, the Torah Shebichtav is quite detailed and elaborate. The Torah is quite clear; there are three tevilot in the Mikvah for the Kohen Gadol. Yet, this clear impression is actually incorrect[ii]; based on the Torah Sheba'al Peh, there are actually five tevilot. So we see here an example which emphasizes the strength of Torah Sheba'al Peh, that what Chazal say becomes our mesorah even if it seems to go against the simple explanation of the Chumash.

So, the Rav explains that how come specifically by the Avodas Yom Kippur we find this example where the Torah Shebichtav is so elaborate and detailed, and still the Torah Sheba'al Peh  presents a different picture. This answer is that Yom Kippur is the Yom Tov of Matan Torah of Torah Sheba'al Peh, and Hashem wants to emphasize the strength of Torah Sheba'al Peh specifically on Yom Kippur. So the parsha of the Avodas Yom Kippur becomes a declaration of the koach of Chazal and the koach of the mesorah. It becomes a declaration that we follow the mesorah even if it's against the simple reading of the Chumash. We always accept the Torah Sheba'al Peh. We should all use this week's Parsha as an opportunity to strengthen our Emunah in and commitment to Mesoras Chazal.

[i] See Rashi in Perek 16- Pesukim 4,23, and 24

 [ii] The implication of the Pesukim is incorrect specifically regarding the Avodah of the Kohen Gadol in the Beis Hamikdash on Yom Kippur. However, there is a concept found in many Sefarim that even when Chazal have a different Mesorah regarding pshat in the Chumash, the simple, literal meaning has significance as well. The Gr"a has a pshat regarding the literal meaning of the Pesukim in Achrei Mos. The Gr"a explains that there were times during the year when the Kohen Gadol would do the Avodah according to the pshat of the Pesukim. The Gr"a's approach is quoted at the end of the Chochmas Adam, and the Meshech Chochmah (Vayikra 16,3) discusses the Gr"a's approach as well.