Perhaps these slides will be helpful. I'll try to explain what happens in your special case.
Let $M$ be a monoid and let $\mathcal{B} M$ be the topos of right $M$-sets. The points of $\mathcal{B} M$ are the left $M$-sets $P$ that satisfy the following conditions:
- $P$ is inhabited.
- Given elements $p_0$ and $p_1$ of $P$, there exist an element $p$ of $P$ and elements $m_0$ and $m_1$ of $M$ such that $m_0 \cdot p = p_0$ and $m_1 \cdot p = p_1$.
- Given an element $p$ of $P$ and elements $m_0$ and $m_1$ of $M$ such that $m_0 \cdot p = m_1 \cdot p$, there exist an element $p'$ of $P$ and an element $m'$ of $M$ such that $m' \cdot p' = p$ and $m_0 m' = m_1 m'$.
For example, the left regular action of $M$ on itself is a point of $\mathcal{B} M$, and as it happens, this point covers all of $\mathcal{B} M$. However, what we need to find is an open cover of $\mathcal{B} M$. The Butz–Moerdijk construction yields such a thing.
Let $K$ be a fixed set of cardinality $\ge \left| M \right|$. An enumeration of $M$ is a partial surjection $K \rightharpoonup M$ with infinite fibres. An isomorphism of enumerations of $M$ is an isomorphism of left $M$-sets making the following diagram commute:
$$\require{AMScd}
\begin{CD}
K @= K \\
@VVV @VVV \\
M @>>> M
\end{CD}$$
Choose a representative in each isomorphism class of enumerations of $M$. We define a groupoid $\mathbb{G}$ as follows:
- The objects are the chosen representatives.
- The morphisms $\alpha \to \beta$ are tuples $(\alpha, \beta, m)$ where $m$ is an invertible element of $M$. (We do not require any compatibility with the partial surjections here.)
- Composition is given by $(\beta, \gamma, m) \circ (\alpha, \beta, n) = (\alpha, \gamma, m n)$.
Write $G_0$ (resp. $G_1$) for the set of objects (resp. morphisms) in $\mathbb{G}$. There is a Galois topology on these making $\mathbb{G}$ a topological groupoid:
- The basic open subsets of $G_0$ are the subsets
$$U_{\vec{i}, C} = \{ \alpha \in G_0 : \alpha (\vec{i}) \in C \}$$
where $\vec{i}$ is an $n$-tuple of elements of $K$ and $C$ is a right $M$-subset of $M^n$.
- The basic open subsets of $G_1$ are the subsets
$$W_{\vec{i}, C, \vec{j}, D} = \{ (\alpha, \beta, m) \in G_1 : \alpha (\vec{i}) \in C, \beta (\vec{j}) \in D, \alpha (\vec{i}) \cdot m = \beta (\vec{j}) \}$$
where $\vec{i}$ and $\vec{j}$ are $n$-tuples of elements of $K$ and $C$ and $D$ are right $M$-subsets of $M^n$.
The theorem of Butz and Moerdijk is that $\mathcal{B} M$ is equivalent to the topos of equivariant sheaves on this topological groupoid $\mathbb{G}$. Note that the domain and codomain maps $G_1 \to G_0$ are locally connected, hence open a fortiori.
The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be represented as $BG$.
I think before Joyal–Tierney's paper it was also known how to generalize from profinite group to general localic groups.
Joyal–Tierney's theorem shows that if you replace "pro-finite group" by "localic groupoid" then you actually get all Grothendieck toposes this way.
You can't directly recover Grothendieck's theorem from Joyal–Tierney's theorem in the sense that the theorem as stated above doesn't tell you for which toposes the localic groupoid can be chosen to be a profinite group. But if you are familiar with the method used in the paper and how the groupoid is obtained (which in my opinion are even more important than the theorem itself) then it is fairly easy to recover Grothendieck's theorem. For example, it immediately follows from Joyal–Tierney's paper that a topos is of the form $BG$ for $G$ a localic groups if and only if it admits a point $* \to \mathcal{T}$ which is an open surjection (which does feel similar to Grothendieck's theorem in terms of a fiber functor).
Regarding the use of internal logic, it is definitely not essential, it just makes everything simpler (at least if you are ok with its use) but one could do without it.
The main way they use internal logic is that in the first sections they prove some results about sup-lattice and frames locales, that are later applied not to sup-lattices and frames, but to sup-lattices and frames in a topos $\mathcal{T}$.
I believe there are also a few places where they make a claim about a morphism of locales $f:X \to Y$ and then only prove it when $Y$ is the point (sorry I don't have the paper with me to give precise reference).
Another place where one can consider they use a bit of internal logic — though this one might be only at the level of intuition — is when they show that every topos admits an open cover by a locale. They do this by considering the topos as a classifying topos of some theory $T$ and considering the propositional theory $T'$ of "enumerated $T$-models", that is, $T$-models that are explicitly given as a subquotient of the natural numbers. Though if I remember correctly, they present the argument in a way that doesn't directly involve any logic… (and in any case there are other proofs of this results that are purely in terms of sites, for example the one in MacLane and Moerdijk's book "Sheaves in geometry and logic").
Best Answer
The groupoid representation of Joyal and Tierney may be identified with the truncated simplicial diagram appearing in the statement of Theorem 2 of the Lurie notes mentioned in the question. That is, a groupoid object is precisely a diagram of the shape indicated in Theorem 2 satisfying some axioms, which are automatically satisfied by construction when one takes pseudopullbacks as indicated in the diagram. In Joyal and Tierney's theorem, the groupoid in localic toposes is precisely this groupoid: it has topos of objects $U$, topos of 1-cells $U \times_X U$, and so forth.
To elaborate a bit, the diagram displayed in Theorem 2, which looks like
$U \times_X U \times_X U ^\to_\to \to U \times_X U ^\to_\to U$
(implicitly there are also some maps going back in the reverse direction) is really just the first few stages of the Cech nerve of the map $U \to X$. The Cech nerve as a whole is indexed by the dual simplex category $\Delta^{op}$, and here we just have the part on the 3 smallest objects of $\Delta^{op}$, namely $[0], [1], [2]$. It may be more familiar that when you truncate all the way to just two objects $[0],[1]$, you get the maps $U \times_X U ^\to_\to U$ -- the kernel pair of the map $U \to X$, which is an equivalence relation. The higher analog of the fact that a kernel pair is always an equivalence relation is that the Cech nerve of a map is always a groupoid object -- $[0]$ corresponds to the objects of the groupoid, $[1]$ corresponds to the morphisms, $[2]$ corresponds to composable pairs of morphisms (not to 2-cells), and in general $[n]$ corresponds to composable $n$-tuples of morphisms.
You can more generally think of an internal category object (not just an internal groupoid object) in a category as a certain type of simplicial object, as described here.