This is very simple, but as far as I can tell it has not been asked yet.
Let the group $G$ act on the set $S$ and define an equivalence relation by $x \sim x'$ if there exists a $g \in G$ for which $gx=x'$.
Proving reflexivity and transitivity is easy, so let's look at the symmetric property: Say $x \sim x'$ with $gx=x'$. Then we have $x=x'g^{-1}$. So $x$ is equal to $x'$ multiplied by an element of $G$, but does this work since we are now using right multiplication? Can we do something 'clever' like $ex=x'g^{-1} \Rightarrow xe=g^{-1}x' \Rightarrow x=g^{-1}x' \Rightarrow x' \sim x$? Something about that last bit seems foul to me.
The group theory tag isn't really appropriate here. I would create a 'group actions' tag if I were able.