The rabbis considered 'shame' (busha) as one of the three components of the Jewish ethico-ethnical makeup. Together with the practice of benevolence (gemilut chasadim) and compassion (rachmanut). The Rabbis went as far as to say that whoever does not posses the trait of shame, could be presumed that "his ancestors did not stand at Mt. Sinai" (an euphemism to say that he might not be ethnically Jewish).
Besides seeing shame as an innate condition of the Jewish character--in complete contradiction with the alleged Jewish 'chuptza'-- the Rabbis explained that 'shame' is a prerequisite for a perfect Teshuba (repentance).
Maimonides writes that when a man or a woman commits a sin... "they shall confess the sin they have done... by saying, `O Lord, I have sinned, transgressed and rebelled in front of You; I have done such- and-such, and I am ashamed of my actions and will never do it again'."
The feeling of shame is a critical step for a sincere process of Teshuba.
Why?
Because unlike guilt--which is a private feeling-- shame consist in the uncomfortable sensation of facing our flaws or misdeeds in front of others.
God is invisible. It is extremely challenging for a Jew to be aware of His constant presence. Therefore, we are not easily ashamed of doing something wrong privately (= "just" in front of God), as we are naturally ashamed of doing something wrong in front of other people.
Now, feeling embarrassed while confessing our transgressions to God would mean that we have achieved this elevated level of closeness to God: The realization and awareness of God's presence. The higher our state of awareness of God's presence, the higher the feeling of shame we feel in front of God, and vise-versa.