Just as Cayley's theorem states that every group is a subgroup of a symmetric group, Yoneda's lemma states that every (locally small) category $C$ embeds into a category of functors defined on $C$.
Specifically, Yoneda's lemma states that if $F:C\to Set$ is an arbitrary set-valued functor, then $F(A) = Nat(\hom(A,-), F)$, so that natural transformations from a hom-set functor are in bijective correspondence with the elements in the functor image.
For this to be a straight generalization, we may consider a group $G$ as a category $C_G$ (actually a groupoid) with a single object $\ast$, and each group element a morphism. Then, there is only a single object, so a set-valued functor is the same thing as a set $S$ with a map from $G$ to $Bij(S,S)$, the set of bijections from $S$ to $S$. We can see this by unrolling the definition of set-valued functor: $\ast$ goes to $S$, and each element of $G$ goes to some map $S\to S$; all of which maps have to be bijections since otherwise the group properties suffer.
Suppose now that we have some such $G$-set $S$, Yoneda's lemma tells us that its elements are bijective with natural transformations from $G$ to $S$; so what is a natural transformation here? Our two functors are $S$, that takes $\ast$ to $S$, and $\hom(\ast,-)$ that takes $\ast$ to $G$; both as sets. A natural transformation of these is a set-valued map from one image to the other, such that the 'obvious' square of induced maps from morphisms in $C_G$ commutes - thus, $Nat(\hom(\ast,-),S)$ is the collection of $G$-set maps from $G$ as a left $G$-representation to $S$ itself.
One of the things Yoneda brings is a bijection $Nat(\hom(a,-),\hom(b,-)) = \hom(b,a)$: the Yoneda embedding. Applied to the group situation, this tells us that $\hom_G(G,G)=G$. Certainly, left multiplication by an element is a group endomorphism; and what this tells us is that these are all there is. This bijection is exactly what is used in the proof of Cayley's theorem on the wikipedia page.