1)
Why did David compose this Mizmor?
1.
Sha'arei Teshuvah (1:23): This is the foundation of the primary aspects of Teshuvah.
2)
Nasan rebuked David for his sin. Why is this called "Mizmor"?
1.
Erech Apayim (1, note 35, citing R. Meir of Parmishlan): Had a coarse person rebuked him, he would not have accepted it. Rather, he heard the Navi's rebuke in a nice way, accepted it and did total Teshuvah.