1)
Who says this?
1.
Rashi: Keneses Yisrael.
2.
Radak (1, 6): This is said on behalf of the Leviyim.
2)
Why does it mention his right hand forgetting?
1.
Radak: It should forget how to play the harp; the right hand moves the harp strings.
2.
Malbim: Just like the right hand is prepared for all man's motions and affairs, so Yerushalayim is the dearness of all his motions and affairs, to the point that it is impossible to forget it, like one cannot forget his right hand.
