In the question we can see the answer... When B'nei Yisrael are on a low level, looking at hard times, non-believers mock and ask "Where is their G-d?!" Therefore on Yom Kippur night, when we are begging for our lives, we ask Hashem to help us for His Own glory. We don't deserve His mercy on our own merit because we are spiritually destitute, we are totally dependant on Him...Who are we?!
Teshuva is such a beautiful gift that Hashem has given us. In human court, we can beg, plead and apologize all we want, but if we damaged someone, the damage has been done, you can't take it back. Ma she ein kein Hakadosh Baruch Hu, with his limitless mercy, gave us the ability to totally erase any wrongdoing we've done! To not take advantage of this opportunity (which we have all day everyday, not just during the Yamim Ne'oraim) would be a real shame. Hashem is just waiting for us to return to Him(Teshuva- the root is shuv; to return). Don't let this amazing opportunity slip by!