A person has many varied experiences in a day, yet somehow one still knows, I am me. One may feel happy or sad, thoughtful or anxious, the states come and go. Experiencing all these different states, one might think the person would feel discontinuous, but that doesn't appear to happen. One still considers oneself an individual.
How is it that one doesn't loose oneself? The inference is that there is something stable behind the individual, something ongoing, something real. The unchanging background of awareness is what Vedantins call it. And the One who knows this is called "Realized."
As the person remains himself, or herself, despite the constant changes, just so, consciousness itself remains one, despite having the experience of billions of different bodies. Not only is this similar to the person's sense of retaining individuality, it is the same thing, for the background of all persons, that which provides the background, is Consciousness itself.
Vedanta establishes that there is no difference between an individual having various passing feelings and thoughts, and consciousness having the experience of billions of bodies. The person passes in an out of feelings and thoughts, and Consciousness knows the passing in and out of experience, of all the bodies.
The experience of the individual, and the experience of Consciousness are One and the same. The only difference is that the individual feels that his body-mind owns the consciousness, whereas, in fact, the individual is only a focal point in consciousness. The mistaken identity is simply ignorance. To "realize," is simply to see through the ignorance.
Body-mind states come and go, yet the person knows himself as one. Consciousness experiences the coming and going of bodies, yet knows itself as One. The coming and going of body-minds does not split consciousness into the many. There is only One in many forms. Realize that and be free.