Having had a few experiences that seemed and felt like I was leaving my body and floating round my house, I can understand why the people who believe in it, do so.
It feels shockingly real, the first few times I became lucid in a dream the shock woke me up. It seems real.
(and going from the waking state into the hypnagogic state is a trippy experience, full of imagery not unlike what you get with hallucinogens, though milder, ofc)
It's just a state of self-induced hypnagogia, imo.
False awakenings are the most similar phenomenon to AP - also the best example of how perfectly the mind can map out a false reality for you, which is hard to tell from the real thing.