Firstly, there is a big difference between abstract concepts and their concrete representations. It is true that we can encode and reason about natural numbers in ZFC, in the specific sense that we can reason about a model of PA (Peano Arithmetic) within ZFC. It does not mean that somehow the abstract concept of $0$ is the empty-set!
Similarly, it is true that we can express notions like "pair" and "function" and so on in terms of set-theoretic definitions in ZFC, but these notions have been around for far far longer than ZFC, and furthermore there are infinitely many viable definitions to 'make concrete' these notions in ZFC. No mathematician actually thinks of the abstract pair $\langle x , y \rangle$ as $\{\{x\},\{x,y\}\}$ or some other concrete representation.
Furthermore, even our standard mathematical notation shows that we conceive of functions in a more fundamental way than is suggested by the standard encoding as a set of input/output pairs.
For more details, see this post about abstract mathematical objects that are not sets, which also mentions urelements (non-sets such as you and me).
Secondly, it is true that from the viewpoint of ZFC, everything is a set (in the sense that given any two objects $x,y$ we can ask whether $x \in y$ or not). But even then, it is not necessarily the case that everything is "just a bunch of nested empty sets"!
The axiom of infinity is the only axiom of ZFC that asserts the (absolute) existence of some set. Every other axiom can be applied only if you already have some set. Now, informally the axiom of infinity says that there exists an inductive set, where a set $S$ is called inductive iff ( $S$ includes the empty set as a member, and is closed under the successor operation ), where the successor of $x$ is $S(x) := x \cup \{x\}$. Note that the axiom does not stipulate that there is a 'minimal' such set, nor what are the members of such a minimal set!
Well, we can use the other axioms of ZFC to construct $N$ to be the intersection of all inductive sets. But there is no way to prove that $N$ only includes as members the empty set and sets obtained by iteratively applying the successor operation. If that seems strange to you, that is unfortunately the way it is.
You see, ZFC does not have the natural numbers as primitive notions, and the axiom of infinity was concocted precisely to enable ZFC to construct a model of PA; we can define addition and multiplication on $N$ and prove that $N$ satisfies PA. But that means that $N$ is what we take to be natural numbers when working in ZFC! Nothing precludes 'our' set-theoretic universe (if it at all exists) from having an $N$ that has more members than those that you can manually write down, namely $0, S(0), S(S(0)), \cdots$, where $0 := \varnothing$.
And the curious thing about this is that ZFC itself knows that the above may happen! ZFC proves that if ZFC is consistent then there is a model of ZFC that has extra (called non-standard) members of $N$.
Finally, most mathematics is in fact independent of set theory, and can be recovered in very weak theories of arithmetic such as ACA, as briefly described here. For real-world applications, it is even better, because there is no evidence of infinitary objects in reality, and there is even a humorous grand conjecture by Harvey Friedman:
Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. (EFA is a fragment of PA.)
