What does the nLab article on ends means exactly with left and right action?
The only way I could make sense of this is that, if $D=\text{Set}$, we have for each morphism $f:c\to c'$ the maps $\{L_f,R_f\}:\prod_{c\in C}F(c,c)\to F(c,c')$ with $L_f:=F(1,f)\circ\pi_c$ and $R_f:=F(f,1)\circ\pi_{c'}$ and the end picks now the subobject of $\prod_{c\in C}F(c,c)$ where those maps coincide.
But I always thought an action would be something mapping back into the object itself, is that not the case?
It might help to understand how group actions (and many other kinds of actions) can be generalized in a categorical setting. First, every group $G$ can be considered as a special kind of category. This category, often notated $\mathbf{B} G$ consists of a single object $\bullet$ and $\hom(\bullet, \bullet)$ is defined to be $G$. The identity for $\bullet$ is the identity in $G$ and composition is the binary operation in $G$. The unit laws and associativity follow from the corresponding facts in $G$.
There are some interesting facts about this construction (for example, Cauchy's embedding theorem for groups is just the Yoneda lemma), but the construction we're interested in is the action of a group. A (left) group action of $G$ is simply a functor $\mathbf{B} G \to Set$. You can check the details, but the gist of it is this: a functor $F: \mathbf{B} G \to Set$ maps $\bullet$ to some set $X$ and for each morphism $g$ in $\mathbf{B} G$ (that is, for each element of $G$), we get a map $F(g): X \to X$. This can be unpacked into a set $X$ and a function $\cdot: G \times X \to X$ (written as infix like $g \cdot x$). Functorality is precisely what's needed to say that $e \cdot x = x$ (because $F(e) = id_X$) and $g \cdot (h \cdot x) = gh \cdot x$ (because $F(g) \circ F(h) = F(gh)$).
This suggests an immediate generalization. An action of a category $C$ on $Set$ is a functor $C \to Set$. Now the action isn't restrained to acting on a single set, but might have a different set for every object in $C$. We could even go so far as to say that an action of a category $C$ on a category $D$ is simply a functor $C \to D$. In this context, the "actions" induced by a functor $F$ are simply the maps $F(f): F(c) \to F(c')$ induced by maps $f: c \to c'$ in $C$. Notice how this corresponds to the actions $g \cdot -: X \to X$ in the case of groups.
A right action can also be described like this via a functor $\mathbf{B} G^{op} \to Set$. More generally, you could even talk about both at once with a functor $\mathbf{B} G^{op} \times \mathbf{B} G \to Set$. Now this is a set with both a right action and a left action on that set. We could even have different groups for the left and right actions with $\mathbf{B} H^{op} \times \mathbf{B} G \to Set$. Generalizing everything, we could consider a left/right action to be a functor $D^{op} \times C \to E$. The left actions of such a functor $F$ are the maps $F(1, f): F(d, c) \to F(d, c')$ for morphishms $f: c \to c'$ in $C$. The right actions are the maps $F(g, 1): F(d', c) \to F(d, c)$ for morphishms $g: d \to d'$ in $D$.