Let $k$ be some field, say of characteristic 0, and let $G$ be a finite group considered as a discrete algebraic group. Then we get the classifying Deligne-Mumford stack $BG$ and its etale presentation $p: \mathrm{Spec}\ k\to BG$. We also have the structure map $\pi: BG\to \mathrm{Spec}\ k.$
It is possible to identify quasicoherent sheaves on $BG$ with $k$-linear representations of $G$. To be honest, I don't completely understand this yet, but the gist is (correct me if I'm wrong) that quasicoherent sheaves over various stacks form a stack themselves, and the descent data needed to descend a sheaf over $\mathrm{Spec}\ k$, which is uniquely defined by its vector space of sections, to $BG$ is precisely the data of a $G$-action.
What I'm completely missing is how under this identification we have $\pi_*(V)=V^G,$ where $V$ is a $G$-representation considered as a sheaf on $BG$. The functor $\pi_*$ should be the same as the global section functor, so we have to show that $\Gamma(BG,V)=V^G.$ We have a cartesian square
$\require{AMScd}$ \begin{CD} G @>s>> \mathrm{Spec}\ k\\ @V t V V @VV p V\\ \mathrm{Spec}\ k @>>p> BG \end{CD} The sections over $\mathrm{Spec}\ k$ is just $V$, the sections over $G$ is $V\times G$, and hence, by definition, the sections over $BG$ should be the equalizer of the maps induced by $s,t$. But aren't $s,t$ the same unique map $G\to \mathrm{Spec}\ k$?