This answer is going to be a bit too informal, but I hope it helps.
Imagine we have the collection of all sets. Let us call them the real sets, and their membership relation the real set membership. The empty set is "actually" empty, and the class of all ordinals is "actually" a proper class.
Now that we have the real sets we can use them as the "ontological substratum" upon which everything else will be built from. And this, of course, includes formal theories and their models.
A model of any first-order theory is then only a real set. This applies to your favorite set theory too. So the models of your set theory are only real sets (but the models don't know it, just as they don't know if their empty sets are actually empty or if their set membership is the real one).
This view fits well, for example, with the idea of moving from a transitive model to a generic extension of it or to one with a constructible universe: we are simply moving from a class of models to another one, each one consisting of real sets.
But this view also leaves us with too many entities, and maybe here we have an opportunity to apply Occam's razor. It looks like we have two kind of theories: one for the real sets, which is made of things that are not sets (we can formalize our informal talk about them, but that does not make essentially any difference), another one for the models of set theory, which is made of sets.
The real sets and the theory of the real sets belong to a world where there are real sets, but there are also pigs and cows, and human languages and many other things. We don't need all that to do mathematics, do we? So why not diving into the wold of the real sets and ignore everything else?
If this story sounds too platonistic, I am sure it must have a formalistic counterpart.
With my question:
How to think like a set (or a model) theorist.
I expected to obtain an official view about all this stuff. I somehow succeeded on this, but as you can see, I'm still working on it.
Here is a related answer to a related question which I also find useful:
Is it necessary that model of theory is a set?