The idea of handling classes "externally" is something like this. Say we want to study ZFC set theory. If we proceeded in a way similar to how we study groups, we would look at structures that satisfy the ZFC axioms, just as group theorists look at structures that satisfy the group axioms. We would be sitting on the "outside", "looking in" at these structures.
If $M$ is a structure that satisfies ZFC, we quickly realize that there are some subsets of $M$ that are represented by elements of $M$, and others that are not. For example, each element of $M$ represents a subset of $M$, but the collection of ordinals in $M$ is not represented by any set in $M$ (because ZFC proves "there is no set containing all the ordinals").
The subsets of $M$ that are not represented by elements of $M$ are the "proper classes" of $M$. Sitting outside $M$, we can manipulate these classes and ask questions about them.
A common question people ask when they arrive at this point in learning set theory is: when we sit outside $M$ and study its classes, what theory are we using? The answer is that we could use any number of theories, including ZFC itself, as our metatheory. Or we could use the same kind of informal reasoning that we use in all other fields of mathematics. We have the freedom to choose any reasonable metatheory in which to study models of ZFC. But we need to separate reasoning done within $M$ from reasoning done about $M$, even if we use ZFC for both the theory of $M$ and for our metatheory.
For various reasons, much of the actual work in set theory is done from the dual "internal" perspective. In group theory, this corresponds to proving facts directly from the group axioms, and then knowing those facts will hold in whatever group we are considering.
From the internal perspective, it is hard to even talk about subsets of the universe that are not represented by sets in our model - because from the internal perspective the sets in are model are all the sets in the universe.
Some set theories, like MK and NBG, allow us to talk about proper classes more directly. ZFC is particularly impoverished in that way, however. Essentially the only way to look for proper classes in ZFC is to look for formulas that define them.
There are several reasons for the focus on internal methods in set theory, which are related to the historical development of the subject and the widespread appeal of the platonistic viewpoint among set theorists.
