What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks?

A category fibered in groupoids over $\mathcal{C}$ is given by a functor $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ satisfying certain conditions (I am not writing the definition as I assume you already know what is a category fibered in groupoids); look at Definition 4.2 in the paper Orbifolds as stacks?

Put a Grothendieck topology on the category $\mathcal{C}$; considering it as a site.

Given an object $U$ of the category $\mathcal{C}$, we consider its fiber; a category, denoted by $\mathcal{F}(U)$, defined as
$$\text{Obj}(\mathcal{F}(U))=\{V\in \text{Obj}(\mathcal{F}):\pi_{\mathcal{F}}(V)=U\},$$
$$\text{Mor}_{\mathcal{F}(U)}(V_1,V_2)=\{(f:V_1\rightarrow V_2)\in \text{Mor}_{\mathcal{F}}(V_1,V_2):\pi_{\mathcal{F}}(f)=1_U\}.$$

Given a cover $\{U_\alpha\rightarrow U\}$ of the object $U$ (remember that we fixed a Grothendieck topology), we consider its descent category, denoted by $\mathcal{F}(\{U_\alpha\rightarrow U\})$.
An object of the category $\mathcal{F}(\{U_\alpha\rightarrow U\})$ is given by the following data:

- for each index $i\in \Lambda$, an object $a_i$ in the category $\mathcal{F}(U_i)$,
- for each pair of indices $i,j\in \Lambda$, an isomorphism $\phi_{ij}:pr_2^*(a_j)\rightarrow pr_1^*(a_i)$ in the category $\mathcal{F}(U_i\times_{U}U_j)$

satisfying appropriate cocycle condition.

Now, given a categroy fibered in groupoids $p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$, an object $U$ of $\mathcal{C}$ and a cover $\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$ of $U$ in $\mathcal{C}$, there is an obvious functor $$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$
(which you might have guessed already but let me say that), at the level of objects $$a\mapsto ((a|_{U_\alpha}),(\phi_{ij}))$$

A category fibered in groupoids
$p_{\mathcal{F}}:\mathcal{F}\rightarrow \mathcal{C}$ is said to be a
stack if, for each object $U$ of $\mathcal{C}$ and for each cover
$\mathcal{U}(U)=\{U_\alpha\rightarrow U\}$, the functor
$$p_{\mathcal{U}(U)}:\mathcal{F}(U)\rightarrow \mathcal{F}(\{U_\alpha\rightarrow U\})$$ is an equivalence of categories.

Now, you can ask what does equivalence of categories has anything to do with "sheaf like" properties? For a functor $\mathcal{D}\rightarrow \mathcal{C}$ to be an equivalence of categories, along other things, for each object $d\in \mathcal{D}$ we need an object $c\in \mathcal{C}$ such that, there is an isomorphism $F(c)\rightarrow d$.

Let $((a_\alpha),\{\phi_{\alpha\beta}\})$ be an object of $\mathcal{F}(\{U_\alpha\rightarrow U\})$. For this, by equivalence of categories, gives an element $a\in \mathcal{F}(U)$ that maps to $((a_\alpha),\{\phi_{\alpha\beta}\})$. That is, given an object $U$ of $\mathcal{C}$, an open cover $\{U_\alpha\rightarrow U\}$, for each collection of objects $\{a_\alpha\in \mathcal{F}(U_\alpha)\}$ that are compatible in some sense, there exists an object $a\in \mathcal{F}(U)$, such that, under appropriate restriction of $a$, you get the objects $a_{\alpha}$. This should remind the notion of sheaf on a topological space. This is how a stack is seen as a generalization of sheaf.

References :

- Notes on Grothendieck topologies, fibered categories and descent theory.
- Orbifolds as stacks?
- How is a Stack the generalisation of a sheaf from a 2-category point of view?