I've been thinking about this quite a bit lately, and I realized something.
You only exist in the present, right?
As in, you only exist right now. You can connect to your past through memories and to the future through predictive reasoning, but you only exist in the present moment.
If you suddenly ceased to exist and were replaced by an exact copy of yourself, 'you' would probably not notice anything.
This connects to death through the idea that your consciousness is only a product of the present moment of your brain's context. Every single one of your past selves died, and all your future selves are unborn. This has lead me to believe that when you die, it's the same as before being born, but it's also the same as whicheevr moment in your life you don't remember.
I really would like to be able to store all the data that makes me me (down to the fundamental particle level) so I can achieve immortality post-mortem. I know it's currently impossible, but I have faith in technology. If I get Alzheimer's before that technology exists and my memories are destroyed irreversibly, I will have failed at acquiring immortality from my perspective.
TL;DR: You only exist right now, your memories make you you, and you die because the 'hardware' storing your memories deteriorates. An exact copy of yourself is as close to the real you as you are.