What does it mean to say a diagram commutes?

Solution 1:

"The diagram commutes" means exactly what it always means: that the map produced by following any path through the diagram is the same.

The notes you are reading made a mistake and forgot to require that $G$ be smooth.

Solution 2:

I think part of the confusion is that the sentence below the definition - "In the above diagram, $G := \Psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \Psi(V)$" - is not meant to be part of the definition, rather it is a clarification. The definition given is that $F$ is smooth iff there exists $G$ (which, from its position in the diagram, is a morphism between open subsets of $\mathbb{R^s}$, that is, is a smooth map*) such that

$\begin{CD} M @> F > > N\\ @V \varphi V V @V V\psi V\\ \varphi(U) @> G > > \psi(V) \end{CD}$

commutes.

*- but I agree this should have been stated explicitly

Does that make things clearer at all?