From a philosophical point of view, it’s not just modern mystics and the enlightened that claim to be one with everything, it was also an ancient stoic belief too.
The simplest way to understand it from a materialistic point of view is that the big bang started billions of years ago and is still ongoing, all matter originated together and at the same time, so there is nothing new within the universe, all of the subatomic particles that make up your atoms are all the same age. It’s just the combinations of those particles that change with time.
If you could zoom out and see the entire universe it would be an atomic cloud of varying densities from place to place but nevertheless connected everywhere, some areas would be thin and wispy others not so much, but nevertheless it is instantly recognisable as ‘one thing.’
This usefully illustrates the point that enlightened people make that there is only one thing in existence and that is the entire universe, all divisions of matter within it are arbitrary and can only be used to describe forms or patterns within the matter of the universe, even when those patterns are independent enough to take on a life of their own.
If you zoom back in again to the level of the atom it’s impossible to see exactly where your atoms end and the rest of the universe’s begin. There’s no clear lines of distinction that exist as anything other than mental constructs that we all agree to as common fields of reference.
Everyday you gain more atoms and also lose some, you cannot keep the atoms that you have, because they don’t belong to you, they belong to the universe.
This makes you a self organising pattern of matter within the atomic cloud that we recognise as the universe, your atoms are locked into a slowly changing pattern of matter that you call an identity, the ‘self,’ that clings to this identity is an illusion because there is nothing else but the universe.
There’s something a little bit more mystical