Show this by induction on the length of the chain of inference. Pairing produces a set that is countable, in fact it has two elements. Comprehension and replacement produce sets that are smaller than preexisting sets, hence countable. Finally, the countable union of countable sets embeds in the integers cross the integers. Think of element y in set x as the fraction x/y. We already showed that fractions are countable, hence the union of countable sets remains countable.
If you want larger sets, such as the reals, you need the power set axiom. And with the power set axiom in hand, sets keep getting larger and larger. See the next page.