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$\DeclareMathOperator\Spec{Spec}$In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are (my conventions are that $X/G$ is the categorical quotient and $X//G$ is the stacky quotient). By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

$\DeclareMathOperator\Spec{Spec}$ If $G$ is an affine group scheme acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are (my conventions are that $X/G$ is the categorical quotient and $X//G$ is the stacky quotient). By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

$\DeclareMathOperator\Spec{Spec}$In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are. By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

$\DeclareMathOperator\Spec{Spec}$In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are (my conventions are that $X/G$ is the categorical quotient and $X//G$ is the stacky quotient). By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\text{Spec } K \to X//G$ of the stacky quotient are. By definition they consist of pairs of a $G$-torsor $P \to \text{Spec } K$ over $\text{Spec } K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading this paper correctly), $G$ is the only $G$-torsor over $\text{Spec } K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

$\DeclareMathOperator\Spec{Spec}$In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are. By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)