Let me write $\def\B{\mathbf{B}}\B G$ for what you call $\mathbb{G}$.
You can check that presheaves on $\B G$ are precisely the right $G$-sets: the unique point of $\B G$ is sent to some set $X$, and functoriality defines a group homomorphism $\def\op{\mathrm{op}}G^\op\to\operatorname{Aut}(X)$.
In particular, the (unique) representable functor corresponds to the $G$-set $G$ itself with the action given by right multiplication.
The Yoneda Lemma in this setting then says, for any $G$-set $X$, that elements of $X$ correspond naturally to $G$-equivariant maps $G\to X$: send $x\in X$ to the map $g\mapsto x.g$, and send a function $f:G\to X$ to $f(1_G)$.