When you awaken from a dream, if done naturally, it's a slow process where your awareness swims upwards to more and more conscious awareness that what you're percieving is not the whole truth of your current reality. Eventually you get to the point where you realize you've been dreaming, or are awakening, and you surface more swiftly back into your waking world, who and what you are here-and-now, as opposed to what you may have thought you were in the dreamstate.

This imagery leads to some of the often-quoted stages of Awakening: snoozing, startled awake, falling asleep.

Another way of talking about Awakening to otherkin is to consider the parallel of 'coming-out-to-oneself' as whatever. You're coming out to yourself as non-human. You've probably known it for a long time on some level, but for the first time, you're really admitting to it as a part of your own identity. In that way, it closely parallels the process of realization that you are gay, bi, etc.