There is a din of piyus on erev Y"K, that someone who did wrong has to go and ask mechila from the person they harmed. The gemara (Yoma 87a) has an additional chiddush. The gemara relates that when R' Zeira knew someone had wronged him, he would make sure to present himself to that person to afford them the opportunity to ask mechila. There was a butcher who insulted Rav and never came to ask mechila, so on erev Y"K Rav went to the butcher himself so that the butcher could ask mechila of him. Even though Rav and R' Zeira were the aggrieved party, they were the ones who went to seek out the chotei rather than wait for the chotei to come to them.
What's the source for this halacha? Why should the aggrieved party have to do anything?
R' Yitzchak Blazer answers that the source is midosov shel haKB"H. Dirshu Hashem b'himatz'o -- Hashem comes close to us during the 10 days of teshuvah to make it easier for us to repent. The King is in the field, as the Baal haTanya teaches; he has made himself readily accessible. Mah hu, af atah. Instead of waiting for the party who has wronged you to step forward, you can take the first step and come to them to afford them the opportunity to ask for forgiveness.
It's a beautiful vort, but R' Wahrman asks a simple question: Since Rav and R' Zeira were going to forgive the other party anyway, why play this game of trying to get them to ask for mechila? Why not just forgive them!?
We see from here, writes Rav Wahrman, that piyus is not just a means to get the other person to forgive. Rav and R' Zeira would have forgiven anyway. But that's not enough. In order to achieve kaparah the chotei does not just needed forgiveness, but he/she needs to go through the process of asking for reconciliation and wanting that forgiveness. It is the sometimes painful and embarrassing experience of making amends which is what leads to true reconciliation -- not just saying, "I'm mochel you."
Let's say that Reuvain and Shimon have a longstanding feud going on, like the Hatfields and the McCoys. Let's be real -- an simple apology on erev Y"K is not going to wipe everything wrapped up in that feud away. So what is this halacha of piyus? What are we trying to accomplish? The answer, I think, is like R' Wahrman suggests: piyus is an expression of the desire to be forgiven. That's the first step, the hardest step. From there, mitzvah gorreres mitzvah and the rest will follow in time.
It's an avodah to forgive people, but it's also an avodah to recognize one's mistakes and to want to be forgiven, to ask to be forgiven.
The goal of all this is to being people together, esp as we approach Y"K. Radomsker in Tiferes Shlomo notes that when the Torah talks about the avodah of Y"K, even though it's a mitzvah l'doros, the parsha does not talk about the avodah being done by "a kohen," or "the kohen gadol", but rather it refers again and again to avodah being done by Aharon haKohen specifically. (It could be that there is a difference between the avodah as done by Aharon and the avodah l'doros - see how GR"A and Netziv following in his footsteps learn the peshut shel mikra.) Perhaps the idea here is that it's not a din in the position of K"G which qualifies one for the job on Y"K, but its embodying the traits of Aharon, of being "oheiv shalom v'rodef shalom..." which qualifies one for the job.
Gmar chasima tovah