There isn't one - it's an axiom that I simply accept for granted. I don't know what it actually means, or whether it really means anything at all; my strong suspicion is that it all ultimately boils down to "I feel being me", which is rather useless.
Is it synthetic a la Kant? Or is “conscious” a part of the conception “I”?
The latter, I think. More precisely, it's the same conception, just different perspectives on it.