In The Kite Runner, what were Amir's sin and his redemption?I would like to know how to explain the author's attempt to show the readers how he...

You have placed your finger on the key theme of the novel.
It is of course Amir's failure to protect his friend, Hassan (who also turns out to be
his half-brother) from being raped that is the driving conflict of the story. Following
this incident Amir feels so ashamed about what he didn't do that he "organises" for
Hassan and his father to leave his household by planting stolen goods on Hassan. This is
Amir's "sin", if you want to call it that. However, it is later on in his life, once he
has left Afghanistan and married in the United States, establishing himself as a writer,
that he is contacted by an old family friend, and feels he needs to try to redeem
himself by helping Hassan. He finds out about Hassan's death for his continued loyalty
to Amir's family, and also discovers the truth of Hassan's parentage. Then in his
attempt to rescue Hassan's son from the same man that raped Hassan, he is very badly
beaten up, but manages to leave to the United States where he and his wife adopt
Hassan's son. This is how Amir finds his repentance, but you would do well to think
about how this theme is much wider than just Amir.