why is amir willing to risk his life in order to save sohrab

He feels intense guilt over Sohrab's father.

Because hassan was his half brother and he'd abandoned him. Her owed it to Hassan to take care of his son in return for the friend Hassan had been to him....... also to make up for turning his back on Hassan out of guilt.

Amir felt badly over how he treated Hassan (Sohrab's) father.