What was the driving force that compelled Amir to transform into the personality that was hidden within?How can I explain that the "cycle of redemption" existed in The Kite Runner?
Most of the impetus behind Amir's later actions, particularly his return to Afghanistan to try to "make things right," are driven by the guilt which has riven him ever since the day that Hassan was raped after chasing down the kite for Amir. He stood by and watched while his most loyal friend was brutally molested, and that inaction has haunted him throughout the rest of his life, no matter how far or how long he got away from the incident and the place where it happened.
Particularly once he finds out that Hassan was, in fact, his half-brother, Amir is desperate to find out whether he might be able to somehow rectify the wrong he committed or felt he did at the time.
In terms of a redemption cycle, it may be his rescuing of Hassan's son from a possible future of constant abuse that he feels he has finally atoned for his inaction after he is able to bring Sohrab back to the US with him.
I agree with the previous post that it was Amir's own guilt which drove him on his "cycle of redemption" in order to clear his own conscience. With Hassan dead, Amir knew that he could never make up to him personally for the injustices he had inflicted upon his friend (framing him for the birthday thefts and idly watching while Hassan was being raped). However, by rescuing Hassan's son (and his own half-nephew), Sohrab, Amir could satisfy part of the guilt that had eaten at him for decades. He furthered his redemptive cycle by reconnecting with Rahim (Baba's best friend) and helping out the Afghani family who aided him on his return to Baghdad. The cycle completes itself when Amir and Sohrab finally connect themselves--while flying kites in California, just as Amir had done long ago with Sohrab's father.