You might also want to think about the Yoneda Lemma as a statement about functors.
A locally small category $\mathcal{C}$ is embedded by the $\mathrm{hom}$ functor in the category $\mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathrm{set})$. This is called the Yoneda embedding. Thus, the $\mathrm{hom}$ functor is fully faithful (this itself is a corollary of the Yoneda lemma), but is not an equivalence of categoriesin general, because it isn't essentially surjectivein general. Not In other words, not every functor $F$ from $\mathcal{C}^{\mathrm{op}}$ to $\mathrm{set}$ is representable- for example, the empty functor which maps each object in $\mathcal{C}$ to the empty set, is never representable. The problem is that the Yoneda embedding does not commute with colimits. But the Yoneda lemma tells you that every functor $F\in \mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathrm{set})$ becomes representable when extended appropriately. In other words, every functor $F$ from $\mathcal{C}^{\mathrm{op}}$ to $\mathrm{set}$ extends to a functor from $\left(\mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathrm{set})\right)^{\mathrm{op}}$ to $\mathrm{set}$ (this is a special case of the Yoneda extension) which does commute with colimits, and is representable.
So one ``philosophical interpretation'' of the Yoneda lemma is the following:
Every functor $F$ from $\mathcal{C}^{\mathrm{op}}$ to $\mathrm{set}$ can be extended to a representable functor from $\left(\mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathrm{set})\right)^{\mathrm{op}}$ to $\mathrm{set}$.
One reference for this point of view is these notes by Akhil Mathew.
