When generalizing from sheafs on a site to 2-sheafs or stacks, it is useful to first rephrase the descent data for ordinary (pre)sheafs in terms of the Cech nerve of a coverage (e.g. https://ncatlab.org/nlab/show/%C4%8Cech+groupoid): $$\text{Match}(\mathcal{U}, F)\cong[\mathbf{C}^{op}, \mathbf{Grpd}](C(\mathcal{U}), F)$$ where $\mathcal{U}$ is a covering family, $F$ is a presheaf (it takes values in $\mathbf{Grpd}$ after composition with the embedding $\mathbf{Set}\hookrightarrow\mathbf{Grpd}$) and $C(\mathcal{U})$ is the Cech groupoid of $\mathcal{U}$, i.e. the 2-coskeleton of the full Cech nerve.
Then, by replacing $F$ with a 2-presheaf (pseudofunctor) $\mathbf{C}^{op}\rightarrow\mathbf{Cat}$ and $C(\mathcal{U})$ with the 3-(co)skeleton of the Cech nerve we obtain the descent data for $F$.
However, I was wondering about the occurence of the category $\mathbf{Grpd}$ of groupoids in the definition of ordinary matching families and, by extension, $\mathbf{2Grpd}$ in the definition of descent data for stacks. Is there any intuition of why we have to regard our presheafs as taking values in these higher categories instead of the ordinary cosmos $\mathbf{Set}$ (and its higher versions)?
And partially related: The Cech nerve is defined as a simplicial object in $\mathbf{C}$ and hence it technically takes values in $\mathbf{C}$, not in $\mathbf{Set}$ (or higher versions). Do we turn this into a $\mathbf{Set}$-valued functor by extending with a forgetful functor $\mathbf{C}\rightarrow\mathbf{Set}$? (And in general by a functor $\mathbf{C}\rightarrow\mathcal{E}$ when working internal to an arbitrary topos $\mathcal{E}$?) I have seen how people turn the data in the nerve into a ($n$-)groupoid, but I haven't really heard them talk about how this happens in terms of functors.
$\mathbf{Grpd}$ is actually the higher categorical analog of $\mathbf{Set}$. Indeed, recall that a set is a collection of object together with an equality, that is a relation which is
The parentheses above are the categorical interpretation of an equivalence relation and shows that the higer categorical version of $\mathbf{Set}$ is indeed $\mathbf{Grpd}$.
But even without this abstract nonsense, just recall what a stack is : $F$ is a stack if you can glue a family of objects along a descent datum : if you have objects $x_i\in F(U_i)$ defined on $U_i$, and want to glue them, you don't want to assume that $x_i=x_j$ on $U_i\cap U_j$, but rather that there is an isomorphism $x_i\simeq x_j$. These objects, together with the isomorphisms $x_i\simeq x_j$ forms the descent datum. This is where the groupoids appears in the formulation $[\mathbf{C}^{op},\mathbf{Grpd}](C(\mathcal{U}),F)$.