This problem has been understood and resolved by a number of thinkers. The Buddha, lots of buddhist masters, then we also have people like Eckhart Tolle who clearly realise it.

The problem is that consciousness is goes beyond conventional logic and descriptions. Interestingly, if it could be grasped by conventional logic it would lose the characteristics of consciousness.

A similar (possibly related) question is why anything exists at all (including why even the concept of something existing exists!)

Unfortunately, having no intuitive insight, I cannot lay claim to an answer, but then notably other thinkers seem to agree that consciousness lies beyond what can be directly explained anyway.

There is an odd branch of logic which allows contradictory statements, possibly such a tool could be used to formally analyse some of the properties of consciousness.