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?