Meditation is not like falling asleep, though it is easy enough to fall asleep during meditation. That’s one of the pitfalls of the practice: meditation tries to simultaneously focus and quiet the mind, and if we miss the mark the mind can sometimes just turn itself off completely and put us to sleep. It happens.
The deeper we go into meditation, the more awake and aware we feel, and the less subjective the experience seems. Go all the way out to nirvikalpa samadhi — nondual awareness — and the sense of a subjective self doing an activity disappears entirely. One is there and not there, because the distinction between there and not there becomes unimportant, and disappears. That’s impossible to describe, because any description will automatically impose a duality on it.
But as I’ve said before, this is not what the finger is pointing at. Meditation is a practice, not a thing in itself. We don’t meditate to have experiences, as though we’re putting trophies on a shelf. We meditate so we know what a calm, peaceful, unmoved mind feels like, so that as we go out into the world we can bring that calm, peaceful, unmoved mind with us.