“The look in his eyes, as they stared into mine, has never left me.” What does that mean?

This happens at the very end of the story, when the narrator is finally looking at himself in a mirror. This is after he's been rescued, but he's recently spent two weeks with a serious illness--and now he looks like a corpse.

So, that quote means that he will never forget what he looked like when he first took a glimpse of himself after surviving the horrors of the Holocaust as well as his own long illness. In other words, the image of himself as a haunted and emaciated (deathly skinny) version of who he once was will never leave his mind for as long as he lives.

Further, take a look at how the narrator is separating himself from his reflection here: he calls the eyes in his reflection "his eyes" (not "my eyes") which presents them as if they belong to someone else. This means that the quote also reveals how Eliezer has become a completely different person because of his experiences. He may even feel as if he is two separate people at once: both the damaged man in the mirror as well as his own real self trapped inside that suffering being.