First-order logic and NBG I know that NBG set theory can be viewed as an interpretation of the first-order logic with equality. However, I do not really understand how do I get this interpretation. 
As I see it: 
I call variables "classes" and add the class membership relation $\in$. I call a class $x$ a "set" if $\exists y \  (x \in y)$. 
But what do I do with constants? Also, there is an empty set and the universal class, and I do not know how to represent them in the form $\emptyset = \dots$ and $V = \dots$

It's not that NBG is an interpretation of FOL, it's that NBG is a theory in FOL, in the same way that e.g. the theory of rings is a theory in FOL. 
The language of NBG has a single binary relation, "$\in$," and no constants or functions. So there's no need to interpret those (any more than we would demand that a ring understand what "$<$" means, say). As to $\emptyset$ and $V$, those aren't constant symbols, but rather definable elements - e.g. NBG proves that there is a unique $x$ satisfying $\forall y(y\not\in x)$, and a unique $z$ satisfying $\forall w(\exists u(w\in u)\implies w\in z)$. We refer to these as "$\emptyset$" and "$V$," but this is done outside the formal language.
