In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\nu]. \end{equation} Where $ \ \ \ D_\mu = \partial_\mu -ig A_\mu $
Is there a physical interpretation for $[D_{\mu},D_{\nu}]$? Every book I read just gives this as the definition of the gauge tensor, because it gives the maxwell tensor in the abelian case and is Lorentz and Gauge invariant... okey, that are some good reasons but I'd like to know if there is physical motivation to use that expression. Why the strength of a gauge field should be related to the non-commutatibity of the covariant derivatives?
Background: I know QFT, some group theory and a little General Relativity. If you start with fiber bundles and stuff, go slow please.