This quote from The Kite Runner occurs in the pivotal chapter of the novel, chapter 7, in which Amir and Hassan have just won the kite-fighting tournament and Hassan goes to run the second-place kite for Amir. We know from earlier in the novel that Hassan is totally dedicated to Amir (he is his friend but also his servant) and he tells Amir, "For you a thousand times over" before he goes to get the kite. Amir thinks of his victory as crucial to his relationship with his father Baba, and it's likely that Hassan knows this, as well. Amir feels he must have this triumph in order to earn his father's love and attention. With all of this set up, the tragedy that takes place in chapter 7 becomes even more poignant.
After Hassan goes to run the kite, Amir tries to find him, and he eventually does, but what he sees is extremely traumatic. Assef, the neighborhood bully, and his two friends have cornered Hassan in an alley, and Assef insists Hassan give him the kite. Hassan, though, is devoted to Amir and has a strong sense of justice, so he will not hand over the prize. As a result, Assef beats and sexually assaults Hassan in the alley while his friends hold Hassan down. Amir witnesses the whole scene and, probably in shock, does not act to stop the assault. The quote in your question occurs at this point in the story. Amir reflects,
I opened my mouth and almost said something. Almost. The rest of my life might have turned our differently if I had. But I didn't. I just watched. Paralyzed.
Amir says here that he was "Paralyzed" and unable to act, but after this quote he also admits that he was a coward and that some part of him felt this was a necessary sacrifice and that Hassan was willing to suffer for Amir's wellbeing. The quote indicates a turning point in Amir's life: he could have "said something" and his entire life could have been different. However, he does not, and this moment of inaction leads to all that follows: his increasing guilt, his actions to drive Hassan and Ali from the house, and his later journey to redeem himself. This quote encapsulates Amir's regret; as an adult, he probably wishes he could go back and do things differently, but in the moment, the fact is that he abandoned his friend.
This quote and the scene that surrounds it are crucial to the plot and character development of The Kite Runner. We also see here the true impetus for the theme of redemption and forgiveness on which the novel truly focuses.