It may be worth looking at this formula as the infinitesimal limit of an exponentiated relation. We have

$$ \exp(- a^\mu D_\mu) \exp(- b^\nu D_\nu) \exp( a^\mu D_\mu) \exp( b^\nu D_\nu) \approx 1 + a^\mu b^\nu [D_\mu, D_\nu] + \dots $$

Just like $\exp( a^\mu \partial_\mu)$ is an operator of translation in a flat space, $\exp( a^\mu D_\mu)$ is an operator of parallel transport, basically translation in a space with nontrivial connection/geometry. The product of these four exponents is a transport along a rectangle. Therefore we can conclude that the field strength tensors $F_{\mu\nu}/G^a_{\mu\nu}$ are measure of how much parallel transport around a small closed curve changes the object transported.

It may be worth looking at this formula as the infinitesimal limit of an exponentiated relation. We have

$$ \exp(- a^\mu D_\mu) \exp(- b^\nu D_\nu) \exp( a^\mu D_\mu) \exp( b^\nu D_\nu) \approx 1 + a^\mu b^\nu [D_\mu, D_\nu] + \dots $$

Just like $\exp( a^\mu \partial_\mu)$ is an operator of translation in a flat space, $\exp( a^\mu D_\mu)$ is an operator of parallel transport, basically translation in a space with nontrivial connection/geometry. The product of these four exponents is a transport along a parallelogram. Therefore we can conclude that the field strength tensors $F_{\mu\nu}/G^a_{\mu\nu}$ are measure of how much parallel transport around a small closed curve changes the object transported.

This also show that field strength tensors are curvatures of a certain (non-metric) connection. They are the gauge field theory equivalent of Riemann tensor from general relativity.