See my other comment :).
B is the set made of all the sets f(X) where X is a subset of A.
A is a subset of the powerset of F. That means A includes some subsets of F, e.g. X.
Now P(n) is the powerset of the first n natural numbers. So P(n) includes all the possible sets including 1 to n of the first n natural numbers (and the empty set, though not important here).
Now, f is a function that maps any element of f ONE TO ONE to a natural number k.
Then we define f(X), where X={a,b,c,…} (element of A, therefore a subset of F), so f(X)={f(a), f(b),…..} which is a set of natural numbers :)
So subset maximal means, there is some X in A for which f(X) (by definition an element of B) is the biggest element of B by size. They want to differentiate “Maximum size set” from “set with Maximum size element” I think :)