Given a group $G$ acting on a category $QC(X)$, you get a sheaf of quasicoherent categories over $BG$, whose fiber at $pt\rightarrow BG$ is $QC(X)$. Global sections of this sheaf are exactly invariants $QC(X/G)$ with an action of "global functions" $QC(BG)$. See http://arxiv.org/abs/1306.4304 for a reference on quasicoherent sheaves of categories.
For nice groups $G$ (affine algebraic groups of finite type), $BG$ is 1-affine. In other words, the functor of taking global sections is an equivalence between sheaves of categories over $BG$ and $QC(BG)$-modules.
So, to recover $QC(X)$, you have to localize $QC(X/G)$ over $BG$ and take the fiber over $pt\rightarrow BG$. This is simply
$$QC(X) = Vect\otimes_{QC(BG)}QC(X/G),$$
where $QC(BG)\rightarrow Vect$ is the pullback to the basepoint $pt\rightarrow BG$. It acquires an action of $Vect\otimes_{QC(BG)} Vect\cong QC(\Omega BG)\cong QC(G)$. Of course, one can just write $X = pt\times_{BG} X/G$ and use the fact that all stacks are perfect, so $QC(pt\times_{BG} X/G) \cong QC(pt) \otimes_{QC(BG)} QC(X/G)$.
$QC(BG)$ is rigid (since $BG$ is a perfect stack), so instead of looking at an action of $QC(BG)$, by adjunction one can look at a coaction of $QC(BG)^\vee\cong QC(BG)$ and take the "cotensor" product. This is just a totalization of the cobar resolution $QC(X/G)\rightrightarrows QC(BG)\otimes QC(X/G)\rightrightarrows...$
To conclude, an object in $QC(X)$ is an object $\mathcal{F}\in QC(X/G)$ with an identification $\mathcal{O}_{triv}\boxtimes\mathcal{F} \cong (f\times id)_* \mathcal{F}$ and higher homotopies expressing associativity. Here $\mathcal{O}_{triv}$ is the skyscraper sheaf at the trivial bundle (under the identification $QC(BG)\cong\mathcal{O}(G)-comod$ it is $\mathcal{O}(G)$ as a comodule over itself) and $f\times id:X\rightarrow BG\times X$. In other words, $\mathcal{F}$ is an eigensheaf with eigenvalue $\mathcal{O}_{triv}$.
Here is one way to think of the "cobar" presentation. Quasicoherent sheaves on $X$ are the same as sheaves on $X/G$ with an action of $p_*\mathcal{O}_X$ for $p:X\rightarrow X/G$. By base change, $p_*\mathcal{O}_X\cong f^*\mathcal{O}_{triv}$. Finally, $f^*\mathcal{O}_{triv}\otimes\mathcal{F}\rightarrow\mathcal{F}$ is equivalent to a map $\mathcal{O}_{triv}\boxtimes\mathcal{F}\rightarrow (f\times id)_*\mathcal{F}$ on $BG\times X$. Its restriction to $pt\times X\rightarrow BG\times X$ is an isomorphism, hence the map itself is an isomorphism (the forgetful functor $Rep(G)\rightarrow Vect$ reflects isomorphisms).
ADDED (in response to a question in the comments):
An $H$-equivariant sheaf (or a $D$-module, or a local system) on $G/N$ is a sheaf $\mathcal{F}$ on $G/N$ together with an isomorphism $p_1^*\mathcal{F}\cong a^*\mathcal{F}$ for $a: G/N\times H\rightarrow G/N$ and $p_1$ the projection to the first factor. One can also write this as $a^*\mathcal{F}\cong \mathcal{F}\boxtimes \mathcal{O}_H$, i.e. $\mathcal{F}$ is an eigensheaf with eigenvalue $\mathcal{O}_H$. Similarly, given any multiplicative sheaf $\Lambda$ on $H$, you can consider $\Lambda$-twisted $H$-equivariant sheaves on $G/N$: these are sheaves $\mathcal{F}$ together with an isomorphism $a^*\mathcal{F}\cong \mathcal{F}\boxtimes \Lambda$. (A multiplicative sheaf is a sheaf $\Lambda$ on $H$ together with an isomorphism $m^*\Lambda\cong \Lambda\boxtimes \Lambda$ for $m:H\times H\rightarrow H$ the multiplication map.) Multiplicativity is needed to make sense of the associativity conditions on the action. Remark: just as above, you can write this as a cotensor product: $QC(G/N)\otimes^{QC(H)} Vect$ for $Vect\rightarrow QC(H)$ the inclusion of $\Lambda$.
