Stabilizer $G_x$ of a $k$-valued point of an algebraic Stack An algebraic stack or Artin stack is a stack in
groupoids $\mathcal{X}$ over the étale site such that the diagonal
map of $\mathcal{X}$ is representable and there exists a smooth
surjection from (the stack associated to) a scheme to $\mathcal{X}$.
In Wikipedia's article on stacks I found in the excerpt a statement on local structure of algebraic stacks I do not understand:

[…] Given a quasi-separated algebraic stack $\mathcal{X}$ locally of
finite type over an algebraically closed field $k$ whose stabilizers are
affine, and $x \in \mathcal{X} (k)$ a smooth and closed point
with linearly reductive stabilizer group $G_x$, there exists an
etale cover of the GIT quotient […].

My question is what is here the stabilizer $G_x$ of $x$ at all? Recall we did not assume that $\mathcal{X}$ is a quotient stack, therefore it seems to me not to make any sense to speak about a "stabilizer group" of $x \in \mathcal{X} (k)= \operatorname{Hom}(\operatorname{Spec} k, \mathcal{X})$.
The point is that in order to talk about a
stabilizer group $G_x$ of $x$ it is necessary to require the existence of a group $G$ acting on the set $\mathcal{X} (k)$ of $k$-valued points.
But for general algebraic stacks there is no reason that there is no reason that such group $G$  acting on $\mathcal{X} (k) $ such that $G_x \subset G$, right? Could somebody help me to resolve my confusion?
 A: This was getting a little bit long for a comment, so I'll just write it here:
Let $X\simeq S//R$ be an algebraic stack presented by a smooth surjective map $S\to X$ with $S$ a scheme, then $R=S\times_X S$, and the pair of maps $R\rightrightarrows S$ has the canonical structure of a groupoid in algebraic spaces (with the additional structure coming from the higher simplices of the Cech nerve). Choosing a point $x$ in $X$ classified by some Zariski geometric point in $\operatorname{Spec}(k)\to S$, form the following big fibre square
$$
\begin{matrix}
G_x & \to & S\times_X \operatorname{Spec}(k)&\to& \operatorname{Spec}(k)\\
\downarrow&&\downarrow&&\downarrow\\
\operatorname{Spec}(k)\times_X S&\to& R &\to & S\\
\downarrow &&\downarrow&&\downarrow\\
\operatorname{Spec}(k)&\to&S&\to&X
\end{matrix}
$$
In this case, the maps $G_x\to S\times_X \operatorname{Spec}(k)$ and $S\times_X \operatorname{Spec}(k)$ are injective, being pullbacks of injective maps, which gives an injective map $G_x\to R$, including as the literal stabilizer of the point $x\in S$ by the 'action' of $R$, it is including as the subgroupoid of automorphisms fixing $x\in S$.
