Caveat: I am not really educated in mathematical philosophy, nor do I deal extensively in mathematical foundations, so the below is just an account of my perspective as a working model theorist.
It's also a bit of a rant, but the question is sufficiently soft that I think it is unavoidable if you want to provide a reasonable answer.
A precise answer to this question will heavily depend on your philosophical stance (regarding the nature and/or existence of mathematical objects), and I hardly consider myself qualified to give an exposition of the possible answers and their intricacies. Further, it depends on the exact foundation for mathematics you subscribe to. But I will try to give a rough answer of how you might think of it as a working mathematician, without worrying too much about philosophy.
The typical "lazy" attitude to resolve this sort of question (well, besides ignoring it entirely) is to take for granted the platonic ideal of absolute existence of mathematical objects as elements in a fixed universe $V$, which, furthermore, satisfies the axioms of ZFC (or some other fixed set theory you think is sufficiently close to "truth").
Now, this may not be entirely satisfying, because you might ask what it means for $V$ to satisfy the axioms of ZFC. After all, to define what it means for $V$ to satisfy the axioms, you need $V$ to, well, exist, and if $V$ is to satisfy ZFC, then $V\notin V$, and if all mathematical objects are to be members of $V$, it follows that $V$ is not a mathematical object. Thus, asking whether it satisfies the axioms of ZFC is nonsense.
To resolve this, you can say that, well, all mathematical objects are elements of $V$, but $V$ exists as a metamathematical object, and as such, it is a model of ZFC. But this raises further question: where do these metamathematical objects come from, and again, what does satisfaction mean?
One possible answer is to think that we have an even "bigger" model of set theory. At first, it may seem that this does not help us, since we're back where we came from. However, note that this "bigger" model does not need to satisfy all of ZFC. It only needs to satisfy enough for us to be able to check whether $V$ satisfies the axioms of ZFC, and to determine their consequences.
In fact, due to the finitary nature of first order logic, we don't really need this bigger model of set theory: the axioms of ZFC are recursively denumerable (under Gödel numbering), and the notion of proof in FOL is inherently recursive. Thus, in order to describe properties of $V$ (and their consequences), it is actually enough to know a little bit of arithmetic. How much exactly - that is what reverse mathematics deals with, more or less. In this view, we can abandon the idea of the absolute existence of $V$ (and other mathematical objects besides the natural numbers) - we can think of them as sort of "syntactic sugar" for the consequences of the axioms of set theory, at least as far as the consequences of the axioms are concerned.
Alternately, instead of going straight to arithmetic, we can go to something else, like type theory, or some set theory weaker (and thus more obviously intuitively "real") than ZFC.
In any event, unless we commit to some sort of infinite descent, you reach a point where you have to take something for granted, like some basic properties of the arithmetic of the natural numbers. In the end, the most basic foundation comes from our intuition about logic and arithmetic. Depending on your philosphical inclinations, you can believe that this intuition is artificial, as is the mathematical reality, or that this intuition is correct, and the mathematical reality - genuine. Or something else entirely.
In my view, to do mathematics, you don't need to assume any of that. It is sometimes useful, but actually (in my view) nonsensical to say that natural numbers, for example, are sets (even though they are modelled quite nicely by some sets, and it is sometimes useful to identify them with those sets). Rather, the idea of the natural numbers, like the idea of the real numbers, is largely independent from the idea of a set. When doing mathematics, it is helpful to "model" (in a more loose sense) these ideas as sets in order to be able to do with them some things we expect to be able to do with sets (like taking a power set, or fixing a wellordering), but we don't really need to think that these things are sets in a model of ZFC, only that they are sufficiently similar to such elements. This is why it is good that we can construct e.g. the reals as Dedekind cuts of rationals - this shows that there is a set that behaves more or less like the real numbers should.
Bottom line: I believe that most (I would like to say all, but there probably are some oddballs) mathematicians almost never think of real numbers as equivalence classes of Cauchy sequences or Dedekind cuts, nor does he, in his mind, equate the number $0$ with $\emptyset$ - because usually, it is not helpful, and if you think a bit, it is not really true. The things we study in mathematics are not usually built by putting sets inside other sets, but rather, by piling ideas on top of other ideas. The question of foundations may be intriguing, but (for a working mathematician) typically irrelevant, and, to some degree, hopeless.
The most baffling thing about this all, really, is that math works surpisingly well. Perhaps because the intuition is based on the physical reality and brains evolved to interact with it.