The way I understand that equation is in terms of bijective correspondences.

An adjunction, crudely, says that arrows \$$F(a)\rightarrow b\$$ in \$$\mathcal{B}\$$ correspond precisely to arrows \$$a\rightarrow G(b)\$$ in \$$\mathcal{A}\$$ .

Let's write \$$h^*\$$ for the \$$\mathcal{A}\$$-arrow corresponding to \$$h : F(a)\rightarrow b\$$ in \$$\mathcal{B}\$$.

(This is just a more compact way of writing the effect of \$$\alpha\$$, ie \$$h \mapsto h^*\$$ is the component \$$\alpha_{a,b}\$$.)

But actually adjunction is a bit more than that – the correspondence has to be _natural_, ie it plays nicely with composition in both categories.

So if we post-compose \$$h\$$ with some \$$g : b\rightarrow b'\$$ in \$$\mathcal{B}\$$, we want \$$g\circ h\$$ to correspond to \$$G(g)\circ h^*\$$.

Similarly, if we pre-compose \$$h^* \$$ with some \$$f : a'\rightarrow a\$$ in \$$\mathcal{A}\$$, we want \$$h^*\circ f\$$ to be the correspondent of \$$h\circ F(f)\$$.

Putting these together we get \$$(g\circ h\circ F(f))^* = G(g)\circ h^*\circ f\$$, which is what the naturality square says.