The Shulchan Aruch (OC 606:1) notes that Yom Kippur does not provide atonement for sins between one Jew and another unless the perpetrator first conciliates his victim. The Mishna Brura says that while there is always an obligation to seek reconciliation with one's victim, we all know how things go: "I'm really busy today, I'll take care of that tomorrow" Tomorrow turns into tomorrow turns into... "Whoa! Yikes! It's Elul already?!" and then "No way! Yom Kippur is tomorrow! Aaargh!" Erev Yom Kippur therefore, by default, becomes the Day of Conciliation.
[NB: Conciliate is, I believe, the correct translation of "m'fayes". Placate and appease might also work, but those both have negative connotations to me. They sound like I am not really re/building a relationship, but just doing whatever is expedient to get the person off my back.]
Halacha demands three attempts at conciliation. Walking up the the person and saying, "I'm sorry, I'm sorry, I'm sorry. Satisfied?" is probably not even once. Each attempt must be sincere, show additional thought, and deeper insight into what the crime entailed. While forgiveness without being asked certainly helps the perpetrator, the real benefit comes from actually facing your victim with a subdued expression and real concern for the damage you have caused. I find it much harder to ask for forgiveness when I really am in the wrong (which, sigh... is the usual situation). After all, when I am admitting fault in order to take the high road, I get to bolster my self-esteem with "wow! I am such a tzadik". Being really wrong and facing the music, though... well, that's painful (and, of course, the most beneficial for my eternal soul).
The Tur and L'vush (thank you, Dirshu) bring another reason for seeking conciliation on erev Yom Kippur: when Klal Yisrael is at peace with one another, the Satan/Adversary is forced to go before the High Court and announce that the Jewish Nation seems to be composed of angels -- each working in harmony with his fellow, each doing his job and praising his friend. There are two novel dimensions added to the mix according to that reason. First, the conciliation actually should be on erev Yom Kippur; that's not just a default. That's the easy one (it's always easy to be told to procrastinate; especially with something that as unpleasant as admitting guilt).
The second one is not so friendly: In order to truly be at peace, you need to admit fault even for things that caused problems because your friend is unreasonably sensitive about something. Of course we always feel that other people are oversensitive and should just get over it already; but sometimes they really are oversensitive. It doesn't matter, at the end of the day, their feelings are hurt and you caused it. Honestly taking that kind of responsibility for our actions and how they affect others goes a long way toward achieving atonement for each of us individually and collectively.