The point is once you collect all the natural numbers into a set, you can now treat that set as an atomic object like any other and you can do all the things you can do with a set to it. So, for example, you can make a set that has the set of natural numbers as an element, you can construct the powerset of natural numbers, you can make functionals (functions taking functions) of natural number functions.
The radical thing about Cantor's set theory was the combination of sets that can contain sets (with the usual operations from finite set theory) and infinite sets. Each idea alone isn't that big a deal. Finite set theory is a perfectly reasonable thing, powersets included. Having a "type" of natural numbers, is also a reasonable thing, it just states what operations you are allowed to do on things that have that type. In particular, in (simple) type theory, you can't make a function that returns a type itself, whereas in set theory it is a completely valid definition to say: $f(1) = \mathbb{N}; f(2) = \mathbb{Z}$.
So the crucial thing is, in the context of the overall theory of sets, the Axiom of Infinity states that not only do the natural numbers exist (effectively) but that you can hold it in your hands and manipulate the set as a whole like any other. This is what finitists rebel against. They don't have a problem with an "infinitude" of natural numbers (though they would say an "unbounded amount"), but with being able to manipulate that infinitude in the exact same way you would manipulate the finite set: $\{1, 2, 3\}$.