excuse me for not reiterating my thoughts from the question that spawned this one; I’ll just build further on it here
In Buddhism, the question of who we really are assumes a central place because our belief in the continuity of the self is seen as the root of suffering. Because we come to identify so strongly with our bodies, our thoughts, our opinions, our memories, etc., we form attachments to phenomena that are necessarily evanescent, and much of our energy is exhausted in trying to maintain a fixed identity in this world of change. That effort is ultimately doomed to failure anyway.
One way Buddhist practice attacks this problem is by giving the practitioner the task of going and finding that self-essence he’s so convinced he has. In the course of years of looking for that self and coming up empty handed, the practitioner is forced to come to grips with the illusory nature of the self.
That’s more than a philosophical discovery, however, because the constellation of worries, fears, aversions, disappointments, loathings, jealousies, cravings and sundry other forms of suffering that have agglomerated to that illusion of self have now also been seen as groundless.
At that point, one is free to just be, without concern for who it is that’s being. And, as tim already mentioned, one becomes able to transcend the self/other barrier that is at the heart of so many of the world’s troubles.