If you take the entire Psalm in context, you'll see that the writer is being punished by God for some sin of which he is guilty. The situation is described in verses 7-11 (NASB):
And now, Lord, for what do I wait?
My hope is in You.
Deliver me from all my transgressions;
Make me not the reproach of the foolish.
I have become mute, I do not open my mouth,
Because it is You who have done it.
Remove Your plague from me;
Because of the opposition of Your hand I am perishing.
With reproofs You chasten a man for iniquity;
You consume as a moth what is precious to him;
Surely every man is a mere breath.
As you can see, the writer is being "chastened" by God for some "transgression". He is asking God how long must he wait for the reproof to come to an end. The "plague" from God is so bad that it makes him feel like he is close even to death. He describes how God consumes "what is precious to him", so it's possible that the punishment involved the removal of something that he cherished.
Therefore, when in verse 13, he asks God to look away, he is asking God to end his punishment. He wants God to look away from his sin and to forgive him. In the first 6 verses, the writer describes what he has learned from the ordeal. He seems to be saying "see, I've learned my lesson, now please make this stop."
Presuming the author of the Psalm is David, the events described do line up quite well with the death of his first child with Bathsheba. As likely as that seems, however, it is speculation, since the text doesn't specify. Here's the account of his child dying, as judgment for his sin:
2 Sam. 12:15-18 (NASB)
Then the Lord struck the child that Uriah’s widow bore to David, so that he was very sick. David therefore inquired of God for the child; and David fasted and went and lay all night on the ground. The elders of his household stood beside him in order to raise him up from the ground, but he was unwilling and would not eat food with them. Then it happened on the seventh day that the child died. And the servants of David were afraid to tell him that the child was dead, for they said, “Behold, while the child was still alive, we spoke to him and he did not listen to our voice. How then can we tell him that the child is dead, since he might do himself harm!”