What is meant by the term „set“ when talking about models of ZFC? A model $\langle M,R\rangle$ of ZFC is a set $M$ together with a binary relation $R$ on $M$. My question is: what exactly do we mean by saying $M$ is a set, since it somehow comes from „outside“ the context of ZFC.
I’ve seen $M$ described as an „external set“ and the elements of $M$ as „internal sets“, like in the answer by user Qiaochu Yuan on this post: How can there be genuine models of set theory?
He writes:

The basic distinction you need to make is between the external and internal notions of set. Let me take as granted a primitive and unspecified notion of set: this will be our external notion of set. For any first-order theory 
in a language , a model of  is a set (in this external sense) equipped with functions and relations satisfying the appropriate axioms, etc. In particular, a model of, say, ZFC is a set  equipped with a binary relation satisfying etc. etc.

Is it true that these „external sets“ are simply build upon a purely intuitive notion of what it means to be a set?
Is there any way to make this notion of an „external set“ rigorous or at least partly rigorous? And could one say that ZFC is insofar the „basis of external set theory“ as that the axioms of ZFC are at least intuitive rules/guidelines for dealing with external sets?
For example: What does it mean that the underlying set $M$ of a model is countable? (as is assumed in some undecidability proofs)
 A: 
What exactly do we mean by saying $M$ is a set?

We mean exactly the same thing as we mean when we say $\mathbb{N}$ (the set of natural numbers) is a set or $\mathbb{R}$ (the set of real numbers) is a set, or a group is a set equipped with a binary relation satisfying some axioms, etc. etc. The definition of "group" or "ring" or "topological space" all use the word "set", and the definition of "model of ZFC" uses the word "set" in exactly the same way.
The only thing that makes the definition of "model of ZFC" more confusing than these other definitions is that if $(M,\in)$ is a model of ZFC, we think of the elements of $M$ as "sets" and the relation ${\in}\subseteq M^2$ as the "membership relation" between them. It's these elements of $M$ that Qiaochu refers to as "internal sets". The "external sets" are just the ordinary sets of mathematics.

Is it true that these „external sets“ are simply build upon a purely intuitive notion of what it means to be a set? Is there any way to make this notion of an „external set“ rigorous or at least partly rigorous? And could one say that ZFC is insofar the „basis of external set theory“ as that the axioms of ZFC are at least intuitive rules/guidelines for dealing with external sets?

Yes, when we take ZFC to be a foundation of mathematics, we mean that we are going to use the axioms of ZFC to reason about sets ("external sets"). ZFC does not tell us what sets are, it just gives us some rules for reasoning about them. I don't know why you call these rules "intuitive" - they are quite formal.

For example: What does it mean that the underlying set $M$ of a model is countable?

It means exactly the same thing as "countable" always means: There is a bijection between the underlying set of $M$ and the set of natural numbers $\mathbb{N}$. (Now $M$, being a model of ZFC, also has an internal set called "$\mathbb{N}$" and internal notions of functions, bijections, and countability for internal sets. But all of that is irrelevant, since the underlying set of $M$ is just an ordinary set, not an internal set. If there's a possibility of confusion, we would write the internal version of $\mathbb{N}$ as $\mathbb{N}^M$.)
