This perspective seems to be absent so far, even though this is a very old question.

To properly credit the source for this idea, the perspective seems to be taken for granted in MacLane and Moerdijk's *Sheaves in Geometry and Logic*, which was where I was introduced to it.

The observation is the following: Categories are generalizations of monoids, and functors are the correct analog of representations.

**The analogy:**

This goes as follows. If $M$ is a monoid, we can define a one object category, which I'll also denote by $M$ which has a single object $*$ and $M(*,*)=M$ with the composition coming from the multiplication in the monoid.

Then if $M=G$ is a group for example, then a functor $F$ from $G$ to $\newcommand\Set{\mathbf{Set}}\Set$ is a choice of set $X=F(*)$ together with mappings $F(g):X\to X$ for all morphisms $g\in G$. If we write $g\cdot x$ for $F(g)(x)$, then the equation $F(g)\circ F(h) = F(gh)$ becomes $g\cdot (h\cdot x) = (gh)\cdot x$ for all $x\in X$, $g,h\in G$. This is exactly what it means for $X$ to be a $G$-set.

A natural transformation between two functors $F,F' : G\to \Set$ is a map $\phi : F(*)\to F'(*)$ such that
$g\cdot \phi(x) = \phi(g\cdot x)$ for all $g\in G$ and $x\in X$. So morphisms of functors are $G$-equivariant maps, as they should be.

Replacing the category of sets with any category you care to think about gives the expected notion of $G$-representation in that category.

**Back to categories and functors**

In this perspective, a covariant functor $\newcommand\C{\mathcal{C}}F:\C\to \Set$ is a choice of set $F(c)$ for all objects $c\in\C$ and a function $F(f) : F(c)\to F(d)$ for each morphism $f:c\to d$ in $\C$. Subject to the requirement that $F(fg) = F(f)F(g)$ when $f$ and $g$ are composable arrows.

We can think about this as a family of sets $X_c$ for all $c\in \C$ such that for $f:c\to d$, and $x\in X_c$, $f_*x\in X_d$ and $g_*f_*x=(gf)_*x$ when $g$ and $f$ are composable.

A natural transformation between two representations $X_c$, $Y_c$ is a family of maps
$\phi_c:X_c\to Y_c$ such that
$\phi(f_*x)=f_*\phi(x)$ whenever this makes sense.

Now the Yoneda lemma becomes the following observation.
The functor $\C(a,-)$ is "freely generated" as a $\C$-representation by $1_a$.
What I mean by this is that a natural transformation $\C(a,-)\to F-$ is determined by the image of $1_a$ and there are no restrictions on the choice of image (except that of course it must lie in $Fa$).
This is because for any morphism $f$, we have
$$\phi(f)=\phi(f_*1_a)=f_*\phi(1_a).$$

The contravariant version is identical, except now we think of functors as right $\C$-representations because contravariance becomes the rule $x|_f |_g = x|_{fg}$, where $|_f$ denotes the action of $f$ on $x$ when this makes sense.

This philosophical perspective is related to Sridhar Ramesh's answer, based on what I can see, but I'm not really familiar with the algebraic theory perspective, and I think this is a bit more of an elementary algebraic viewpoint.

The point here is that you should think of Yoneda functors $\C(a,-)$ or $\C(-,a)$ as the free objects in a single variable supported at an object $a$.

the “philosophical” meaningof the Yoneda lemma. $\endgroup$