What is "Anah"?
Radak: It is Hoda'ah. Also "Ana Chata ha'Am ha'Zeh" (Shemos 32:31) is Hoda'ah (admission of sin). Here it is thanks for making David His servant.
Malbim: This begins a request to save him from those who rise against him, i.e. Melech Ashur and his multitudes.
Why did he say "Avdecha Ben Amasecha"?
Rashi: The nature of a slave bought in the market is unlike that of one born in the house. (The latter is more subservient. - PF)
Radak: He mentions the mother, for the primary formation of the son is in her womb. He grows from the menstrual blood and is formed there for the set time. Also after he leaves, he is fed from his mother's milk. Therefore, the son's nature is tied to his mother's more than to his father's. This is why it says regarding Malchei Yisrael "the name of his mother", for the son's acts are usually like his mother's. Even though man has choice, and his deeds are based on how he conducts himself and his habits, part is due to the nature of his formation. I am Avdecha in two aspects - I accustomed myself to Your Avodah, and it is in my nature, for my mother was Your slave.
Malbim: I am Your slave - it is not proper that You hand me over to another master.
Why does it say "Pitachta l'Moserai"?
Yevamos 77a: David said to Hashem, two binds that were on me, You opened. You forbade Amonim and Mo'avim, but permitted an Amonis and a Mo'avis. [David descends from Rus the Mo'avis, and Rechavam ben Shlomo is from No'oma the Amonis.
Radak: I knew that my salvation was from You. David was not incarcerated, but sometimes it was as if he was trapped, e.g. at Sela ha'Machlokos, and in front of Achish (Shmuel I, 23:28, 22:14-16).
Malbim: It is proper that You open the binds that Melech Ashur put on me.
