Note that there is one axiom for every such predicate φ; thus, this is an axiom schema.
To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique
My question is why, by the axiom of extensionality, is this set unique? The Axiom of Extensionality is about set equality; what is the relation between the Axiom of Extensionality with uniqueness of this set?