>>10853143I'm not an expert on it, but I think it's roughly like so: For models of ZF you use the big Neumann universes by applying the power operation iteratively and whenever you're before a new limit ordinal you form an arbitrary union to actually get there . Those are models of ZF but don't exist via ZF alone - otherwise there wouldn't be small models of ZF such as .
Contrary to just ZF (or ZFC), if you say there's a Grothendieck universe U for every set, say {}, and the axioms say such an U has all power sets and some unions already inside of U, then you get quite big in U. And then the Grothendieck universe axiom of course still also holds for that new U, and so on. You get the ordinal jumps for free by always going up from U to U and the axiom always grants existence.
I think it's just stronger in producing existence of sets than just power set because the new ordinal you produce the aribtrary union with is also generated in U.
Related
https://ncatlab.org/nlab/show/Grothendieck+universe 
