The question seems to come down to a common confusion: how can a model of ZFC be a set, if we want to use ZFC to study sets? Here we are looking at ZFC as an "object theory" - a theory that we are studying, and whose models we are interested in. There is a concept that many students have that we should study ZFC starting with some theory weaker than ZFC.
The answer is, essentially, that things don't work that way. To study "models", we need to work in a metatheory that already has some concept of set or collection. The "metatheory" here is the theory that we use, as a tool, to study the object theory.
There are many options for such a metatheory. One option would be ZFC itself, except that (by the second incompleteness theorem), ZFC can't prove that there is a model of ZFC. We could use ZFC + "there is a model of ZFC" as our metatheory, or we could use a stronger set theory like Morse-Kelley set theory. These are strong enough to study models of ZFC. There are other options, as well.
If we want to work with a "weak" metatheory, which is not strong enough to talk about models, we can only study provability in ZFC, not models per se.
In much of the actual study of set theory, and many set theory textbooks, the metatheory is left unspecified, which tends to perpetuate the confusion about what metatheory is being used. Sometimes the metatheory is left intentionally informal, based on our naive conceptions of sets and collections. Sometimes, models of ZFC are explicitly studied in the metatheory of ZFC + "there is a model of ZFC".
This is true throughout mathematical logic, though. In the study mathematical logic, we apply mathematical techniques to study logic. This means that we apply the same kind of reasoning - including sets, functions, etc. - that we use to study all other kinds of mathematics like algebra, analysis, etc. Occasionally, we do work in weaker metatheories, but in most cases the metatheory we use is ZFC or stronger.
The general idea is that, by studying ZFC as an object theory, we learn more about the nature of sets. This informs us more about our set-theoretic metatheory. We can then use this new insight about the nature of sets to learn more about ZFC, in a kind of cyclical process that has some analogues in textual analysis.