The why-am-I-me question will haunt you throughout your life. It's one of the most significant questions you can ask. Your daily life will distract you from thinking about it... for a while. But it will always return to bother you, like a stone in your shoe. It will remind you that something is not quite right. Something important.
The Question can be rephrased more precisely as follows: “why is reality being experienced from the perspective of my body instead of another, or from any other perspective?” This perspective seems to persist even though the matter and energy in your body are constantly changing. As a child, I used to wonder if I had been someone else the day before, and switched bodies overnight. With all past memories intact in the brain of the new body, I wouldn’t know my consciousness had switched. Even if the switch occurred every minute, I still wouldn’t know. Of course, this thought experiment assumes that no memories are kept in a non-physical soul of some kind.
Others here have given some interesting answers to this question. I will only say it’s a childhood question many have asked, but science will never answer. Science can't answer it because science assumes there is no "preferred" perspective on the world — “the view from Nowhere”, as Thomas Nagel calls it. That assumption underlies the entire scientific enterprise. But your experience of reality contradicts it. If you’re a conscious being, there certainly is a preferred perspective: yours. That’s simply what’s being observed. It’s as if the universe prefers to see itself through your eyes. This wouldn’t be a problem for an unconscious automaton that doesn’t experience anything. In fact, only a conscious being would even ask the why-am-I-me question.
All knowledge depends on perspective. You can imagine the world as someone else. You do it when you read a book or watch a movie, or when you try to understand someone else. But underneath you’re still the one doing the thinking, learning, and experiencing. The poet Sylvia Plath once asked, "Is there no way out of the mind?" Well, maybe death is the way out; she committed suicide. But in life, all the science, math, philosophy, religion, art, etc you think you know — none of it has ever actually been separated from your perspective, or your point of view. Reality could be vastly different from another perspective.
If perspective is so important, then why your body? It seems so arbitrary. Why not the person next to you, or the dog on the floor, or the fly on the ceiling? Why not someone in another country, or an alien in another galaxy, or a being in another universe? Why not God, if you believe in one? Of all possible conscious beings in all possible universes, why is your perspective so obviously preferred? If you think the answer is some kind of dice roll, then exactly what is being rolled, and how? Or if you think conscious perspectives are fundamental souls of some kind, then why is your consciousness bound to your specific body? Either way, the why-am-I-me question will force you to conclude that science is an incomplete description of reality. Perhaps even woefully incomplete.