Like any other part of a biological system, our consciousness has been molded via natural selection. In humans, any divergences in consciousness as compared to other animals likely serve the purpose of presenting oneself as an individual that is stable as well as predictable, and therefore able to cooperate in the reciprocal altruism scenarios that were essential to survival in the Pleistocene. Although it is often cast as an intractable problem, we all know what it is like to experience consciousness. Relativistic philosophers often ponder whether someone could perceive a color in a different way and not know so, but this is highly improbable. Our three types of cone cells respond to ranges of electromagnetic wavelengths in such a specific way that there is little reason to expect that there might be variation in their function between members of the same species, aside from the few color blind exceptions that we already know of. All of the other environmental inputs that we detect are also transduced through physical processes that leave little room for or suggest potential a benefit due to variation. Not all of these sensory inputs reach the level of consciousness. Those that come close compete for attention in the brain and the few that we are able to focus on or recall past instances of at any given moment are what constitute the experience of consciousness.
I don't see why this is so often branded as a "mystery." Certainly the mechanisms for some of the necessary processes are currently quite undetermined but there is no reason that they will remain so forever. There is also no need to suggest anything unphysical or nonclassical.