I believe awakening occurs in a spiritual context. The user of a Devil Fruit must literally become spiritually awakened by losing their connection to the physical world along with physical desires to "understand" how Devil Fruit abilities can affect things other than their own body.
It's something like that. Yes, it may coincide with death or near-death experiences.
This would be a form of outer-body experience because it's not only their own body they are using as a vessel for energy anymore, but instead things other than their own body. Essentially, it means they are projecting their soul into other things.
When you think of the mind and body catching up with Devil Fruit powers... doesn't it usually go mind/body/spirit? Devil Fruits get attached to the spirit part.
I'm not sure why Big Mom and Kaido never did it. I guess they were just not enlightened.