I have troubles wrapping my mind around this equation.
It is trivial to prove, but I prefer to think about the objects in $[C,\text{Set}]$ as generalized objects over $C^\text{op}$, motivated by the Yoneda embedding and the fact that this category is the colimit completion of $C^\text{op}$.
Is there maybe a different description of the isomorphism in question, not via currying but related to the colimit completion, or otherwise related to picture of generalized objects?
I'm not sure this gets at what you're asking, but you could think of a functor out of $C$ as a $C$-action on a collection of objects in the codomain. Then the isomorphism in question is saying that $C_1$-structures on $C_2$-sets are equivalent to $C_2$-structures on $C_1$-sets. There are then analogues in other contexts than plain presheaf categories. For instance, an ordered group is equivalently a group object in ordered sets or an ordered object in groups. Such statements are much less trivial to abstract and prove in full generality than yours, but perhaps they lend some intuition.