How is Johnson's shortest path weighting function $\hat{w}(u, v) = w(u, v) + h(u) - h(v)$ proven by the triangular inequility?
You're right that you can't (necessarily) apply the triangle inequality to the edges of the original graph, but that's not what's being discussed here.

We know $h(u)$ is, by definition, the shortest path to $u$.  Thus we immediately know that $w(u,v) + h(u) \geq h(v)$ because, if it wasn't true, $h(v)$ would not be the shortest path to $v$.  The author is calling this the "triangle inequality" of the reweighted graph.