This can be likened to a child who has soiled himself, and then runs to his mother and begins hugging her, sitting in her lap. Two things happen then: On one hand, the mother becomes dirty from the child—but at the same time, deep emotions of love are awakened when the child hugs his mother. These feelings of love drown out any annoyance about the filth.
The same takes place when an older son did something misguided and made many mistakes. But he comes to his father and says: “It’s true that I made mistakes, but I realize the error of my ways... I want to change, and I want to learn from your wisdom how to change for the future.” Surely, the father will derive much pleasure from this gesture. Any disappointment caused by his son’s mistakes is drowned out by the feelings of love—for these feelings are much more internal than the surface-deep disappointment.
The same applies to our relationship with HaKadosh Baruch Hu, only on a much, much deeper level—for the connection between us and the Ribbono shel Olam is much deeper than that of a father and mother and their child. And when a person comes with the right approach; that is, he understands that for his part he is unworthy, but he comes because he is a son of Avraham, Yitzchak, and Yaakov, and because he appreciates the great love that HaKadosh Baruch Hu has for us, he effects a great nachas ruach through his tefillah.