Why are all polymorphic functions between functors natural transformations?
I think the easiest thing is to refer you to the famous Theorems for free! paper, which explains the intuition behind the use of the Reynolds parametricity theorem.

Intuitively, in System $\mathrm F$ a function $\alpha$ of type $\forall X. F\ X \rightarrow G\ X$, is so polymorphic, that $X$ can be instantiated by a morphism $f$ rather than just types. When one makes this precise, one sees that an arrow $F\ f\rightarrow G\ f$ can only be a commuting square, which exactly expresses the fact that $\alpha$ is a natural transformation.

The proof itself can be derived from Reynolds' classic result from Types, Abstraction and Parametric Polymorphism, which builds a model of System $\mathrm F$ where every type is interpreted by a relation over programs.