After Peter denied Jesus, he cried remembering Jesus' words. So he cried because he remembered that Jesus knew before hand that he would deny Jesus. His mind returned to Jesus' words which told him that he could not but do evil deeds. That is when Peter found out that he was truly a sinner that needed to be destroyed. There was a sincere repentance in his heart.
Most people rely on themselves.
They believe in their power and will, But it is hard to earn God's mercy while they have this kind of heart. A person who thinks he can do something well is drifting farther apart from God. But a person who finds out that he is a sinner who should be condemned is the one who is returning to God. He is the one who will receive God's grace.