Following the exposition in Roman Kossak, Mathematical Logic : On Numbers, Sets, Structures, and Symmetry (Springer, 2018).
If we start with a finite set, with n elements (n little), it is easy to list all its subsets (they are 2^n).
If we use a bigger one (but still finite), like e.g. a chessboard, we can exercise ourselves in selecting different "configurations" of squares : the obvious ones, like that of all black squares, but also every selection of squares (not necessarily adiacent) will do.
Having 64 squares available, we have a quite huge choice of selections.
Each selections is a subset of the set of squares making the chessboard.
Having said that, what is the "philosophical status" of the statement: “There exists the set of all subsets of the chessboard” ?
If we imagine to enlarge the chessboard to a 16 x 16 schema (we have only doubled the size of the side !), the number of possible selections grows up to 2^(256), that is quite huge : around the number of atoms in the universe.
Thus, if the set of all subsets exists, where does it exist?
But here is the key point of the mathematical theory of sets: set theory is the theory of mathematical infinite.
Mathematics lives in an infinite universe. Thus, the limitation discussed above is not consistent with the "infinitistic attitude" of modern mathematics.
Going back to our chessboard, we can freely click here and there to select any set of squares.
It is hard to say that some subsets of the chessboard do exist, but some others don’t. Since none can be excluded, we are inclined to accept that they all somehow "exist", and therefore there must "exist" the set of all of them.
This is the "conceptual" ground of the power set axiom.
Obviously, there is no way to "prove" the axiom; outside mathematics, IMO there is no compelling reason to "believe it", but there is no compelling reason to believe the mathematical theory of sets, either.
From a more "pragmatic" point of view, all "initial" axioms of set theory allows us to assert the existence of very few sets : the emptys set and other sets quite "slim".
To do mathematics, we need the set of naturals, and thus we have to postulate the existence of an infinite set, and that of the reals, and thus we have to postulate the existence of the set of all subsets of the set of naturals.
Again, we need them if we want to use set theory for its primary purpose : to develop the mathematics of infinite.