Axiomatic set theory does not actually solve the philosophical problems with naive set theory.
Set theory started historically with the work of Dedekind, who found ways of witnessing irrational numbers as sets of rational numbers, which by this point has been generalised to a derivation of all mathematical objects as sets of sets of sets, essentially building everything out of the empty set, sets containing the empty set, sets containing those sets, etc. In these constructions it is understood that an infinite set is more precisely defined by its criteria of membership; a set is actually a predicate. If a set is just a predicate, and sets are taken to only include other sets, then what we are really talking about is predicates that may or may not be satisfied by other predicates. You could call such an object a recursive predicate, or a recursive set.
It is well known at this point that it immediately leads to contradiction to even define a kind of object that is capable of representing any predicate over that kind of object, but people really liked the novelty and simplicity of being able to build anything as a recursive set, so the solution that is now accepted is to build a system of axioms that, if true, would behave like recursive sets, but without leading to a contradiction. Wishful thinking is literally taken as the method.
In practice the system people are dealing with is a system of formal
statements, things that you can write down that would be true, if the
axioms of the system were found to apply to anything that exists. These
statements can be turned into sets, but those sets would be sets of other
statements in the system, which is a little bit like having a recursive set,
although slightly less satisfying. These sets of statements might even satisfy
the axioms themselves, (for example the set of no statements is like the empty
set) which is kind of like how model theory is done, but people don't tend to
think about set theory as talking about something circular like that... They
tend to just pretend that recursive sets do actually exist, exactly the way
that naive set theory treated them, but that axiomatic set theory can tell you
what their properties actually are, with each collection of axioms acting as a
different universe
of sets. This is a slightly strange way of
thinking about mathematics, especially when the mathematicians who thought this
way have accrued so much prestige that if an object is constructed from these
axioms, then it would be considered more definite, more convincing that it is
talking about real objects, than if it were described using everyday ideas
and experiences, even if they were the first-principles of everyday experience.
(Not that people really try to do that.)
If you want to understand strange results of set theory, or model theory, I recommend first interpreting them as results about sets of written statements, second as purely hypothetical statements that might be talking about something unknown, and only third as a real world full of all of the accepted objects of mathematics. It is only the interpretation that I claim is contentious; all of the results they give are true, but are true results of written words that don't say much about anything except themselves. If you want to understand what a set really is, first work out what it actually is you want to take sets of, and if the things you are taking sets of are just written symbols like '1' or '25', then that is okay. A set is a bunch of objects that meet some criteria, visualised as a single object. It works fine without axioms, as long as you don't try to reason about a recursive class of sets that simply contain any other set in that class.