"Philosophical" meaning of the Yoneda Lemma

In his Algebraic Geometry class a few years back, Ravi Vakil explained Yoneda's lemma like this: You work at a particle accelerator. You want to understand some particle. All you can do are throw other particles at it and see what happens. If you understand how your mystery particle responds to all possible test particles at all possible test energies, then you know everything there is to know about your mystery particle.

One way to look at it is this:

for $C$ a category, one wants to look at presheaves on $C$ as being "generalized objects modeled on $C$" in the sense that these are objects that have a sensible rule for how to map objects of $C$ into them. You can "probe" them by test objects in $C$.

For that interpretation to be consistent, it must be true that some $X$ in $C$ regarded as just an object of $C$ or regarded as a generalized object is the same thing. Otherwise it is inconsistent to say that presheaves on $C$ are generalized objects on $C$.

The Yoneda lemma ensures precisely that this is the case.

I wrote up a more detailed expository version of this story at motvation for sheaves, cohomology and higher stacks.

If you have basic experience with abstract algebra, the ideas in the Yoneda lemma should be quite familiar and even intuitive; the apparent difficulty is only in recognizing them in this new presentation.

You can think of "category" as meaning the same thing as "algebraic theory in a multisorted language with only unary functions"—the objects of the category being the sorts of the language, the morphisms being the definable functions, and the equalities between (composites of) morphisms being the laws of the theory. From this perspective, a functor from $C$ to $\mathrm{Set}$ is simply a model of the theory corresponding to $C$, and natural transformations of such functors are homomorphisms of models. The Yoneda lemma then is about free models. Specifically, it says that for every sort $s$, the "term model" of terms with a single variable, of sort $s$ (equivalently definable functions with domain $s$) is the free model on a single generator of sort $s$. It may be unfamiliar when expressed as "$\mathrm{Nat}(\mathrm{Hom}(s, {-}), M) \cong M(s)$ naturally in $M$", but that is indeed all this categorical expression is saying

The so-called co-Yoneda lemma mentioned in the other comments also has a nice interpretation from this perspective, amounting to the demonstration that every model can be specified by generators and relations.

I wouldn't say this is The One Right Way to think about the Yoneda lemma, because it's useful to view it from many different perspectives, but this is certainly One Right Way to think about the Yoneda lemma.