Naively speaking, you might expect that every collection you can define exists. But existence in standard set theoretic contexts means being a set.
Russell, with his eponymous paradox, showed that not every collection is a set. The reaction to that was to formulate some axioms and state that things you can derive to exist from these axioms will be sets, and things you can prove to not be sets, are not sets.
Naively speaking there is no reason for the collection of successors or collection of natural numbers to be sets, even if individually we can prove each natural number exists. Other than "we really want them to be" anyway. So this was formalized into the axiom of infinity that states that this is indeed a set.
You could argue that this is not justified to do just that. But in effect this is necessary for modern mathematics, and we know that this axiom gives us significant power, so other than giving it some philosophical justification, there is no way to derive it naively.
Finally, there are many many ways to formulate this axiom. We can require different kinds of sets to exist, and from them to prove the existence of the set of all natural numbers.