Perhaps I am the only one, but I could not help but reflect on what we were actually doing when we invested many hours in Elul and Tishrei asking G-d to forgive us. Between the avinu malkenus (our Father, our King), the selichot (penitential prayers) and the vidduim (confessions), it was amazing just how many times we said various permutations of the same basic request.
As with the purpose of prayer in general, begging G-d to forgive us does not seem to make a lot of sense: He knows exactly what to do, and certainly does not need our requests in order to do it. In fact, much has been written on the paradoxical nature of prayer, even leaving one seemingly exasperated Jewish philosopher, the great R. Yosef Albo (Ikarim 4:16) to conclude with the very philosophical rejoinder of "What can I tell you, it works" (my paraphrase).
So too it is with repentance. We know what we need to do, and no doubt G-d will forgive us if we do it. However if we do not do it, as our Sages (Ta'anit 16) beautifully illustrate, our calls to G-d are as if we are holding on to an impure animal while purifying ourselves in a mikveh – such a purification obviously does not work. If we have no intention of repenting, our cries to G-d are empty and of little effect. But then, why do we cry and plead in shul as if G-d's decision should depend on how loud or often we have cried out for forgiveness?
While R. Albo's answer is not the only one, and many other fascinating approaches to prayer have been put forward, I would like to suggest a possible insight from the East.
I know very little about the East, but in college I briefly studied what seemed to be the most philosophical of martial arts, Aikido. With the risk of doing to Aikido what Rabbi Wein once said could be done to physics, by summarizing it all under the song "Twinkle, Twinkle Little Star," I was able to cull some very profound wisdom from the teachings of this martial art, that has stayed with me until today. The underlying principle of Aikido is to go with someone else's force as opposed to trying to use one's own force, defensively or offensively. In other words, rather than striking one's opponent, one should neutralize him, by pulling him towards the direction of his own blows, thereby going with the preexisting flow of power.
Similar ideas actually exist in Rav Kook's explanation of what he calls "natural Teshuva" and also in the Maharal's explanation of the law of false witnesses (edim zommemim). Yet, I find the idea most graphically and simply explained in the practice of Aikido.
It would appear that our repentance creates the Divine energy to forgive us, and somehow that energy is further brought down by our prayers. Our prayers take the energy that is already in motion and pull it forward into its most complete form. As with Aikido, our prayers for forgiveness allow us to maximize the benefit of a preexisting force outside of ourselves. And also, as in Aikido, without the preexisting force nothing exists. If we are not worthy of forgiveness, there is nothing for prayer to pull out
We are forgiven either way, but when we add prayer to our repentance, it gives us a partnership in the actual Divine forgiveness itself, going beyond the repentance that elicits it. Through our partial ownership of our own forgiveness, we are able to internalize it and appreciate it better. (The Sages allude to this concept, when they say that a person would rather have much less of his own produce than the produce of someone else.)
Allowing us to have a share in G-d's actions is how He creates partnership with the Jewish people. What is true of prayer is even truer of Torah study: The Torah could have been given as a closed book. Instead, it was created as an open book, requiring interpretation, so as to invite us into a holy partnership with Him. As least as much a paradox as prayer, the creation of partnership between man and G-d is a central motif in Judaism.
The partnership that we build over the High Holidays is also what allows us to be ready for Sukkot. Since the Sukkah is G-d's dwelling, we can only live in it when we have become his partners. Otherwise we can only be His guests.
Thus, it is only after the prayer of a Yom Kippur that we can have the audacity to be at home in G-d's dwelling. If so, the true test of our repentance may well be how successful we are at making our Sukkah into our home.