From what I’ve read and understood (Halmos’ Naive Set Theory and this SE post), I reckon that the classes were “invented” as something which won’t belong to any other class, so as to save Set Theory from Cantor’s Paradox, Russell’s Paradox, etc.
Correct?
Formally speaking in $\sf ZFC$ classes are not objects of the theory, but rather of the meta-theory. In other words, classes are just shorthand for formulas which define a collection of sets (with given parameters). Sets, on the other hands, are the objects of the theory.
Every set is a class, since if $A$ is a set, then the formula $\varphi(x,A)$ given by $x\in A$ defines a class which is exactly $A$ when given the parameter $A$.
But what about $\in$? Well, $\in$ is defined for the objects of the theory, namely sets. Therefore if we say $x\in y$, then in principle we say that $x$ and $y$ are sets. But we often abuse notation and allow $y$ to be a class, so if $y$ is a class defined by $\varphi(x)$, then we write $x\in y$ to denote $\varphi(x)$.
So.... how does that all tie in together? Why shouldn't we allow $x$ to be a class as well? Well. We do allow it to be a class. It's just that if $x\in y$, then in any case $y$ is a collection of sets. Therefore $x$ has to be a set. Not to mention that we can't quite assign a variable a formula. So if you want to write $\varphi(x)$, and $x$ is a class given by $\psi$, what we really want to say is that $\exists z(z\text{ is defined by }\psi\land\varphi(z))$. But this again necessitates that $z$ is a set.
This whole thing seeped through when we defined "class-set theories" like Kelley–Morse or Gödel–Bernays, where classes are actual objects. In those theories sets are usually defined as those classes which are elements of other classes, which puts us again in the same place.
Classes are collections of objects, which in set theory traditionally means "collections of sets" because all the objects are sets. So if you are an element in a collection of sets, you are a set.
