“I understood the nature of my curse: I was going to get away with it.”
Amir makes this statement to himself not long after witnessing Hassan get raped. It's clear from his words that he's not just traumatized by what he saw, but also by his failure to intervene. Amir could've prevented what happened to Hassan, but he chose not to, mainly because he was jealous of Hassan for supplanting Amir in his father's affections.
Amir knows that he did the wrong thing, and that he'll have to live with the consequences of his inaction for the rest of his life. As he lies awake in the silent darkness he says out loud "I watched Hassan get raped" in the hope that someone might hear him. The burden of this appalling secret is simply too much for Amir to bear and he wants someone—anyone—to hear it. But no one does. Baba is sound asleep as Amir admits to himself what he did, and so Amir is destined to keep this terrible secret bottled up inside him, possibly for the rest of his life.
Amir interprets this as some kind of curse, the curse of getting away with something he'd rather not get away with. Racked with guilt, Amir would dearly love to turn the clock back to the time before the terrible events of that winter's day. But he can't. And so, unable to unburden his soul to anyone, he's somehow going to have to live with the pain of this dark secret.