For the abelian QED theory, $$[D_\mu, D_\nu]=ieQF_{\mu\nu}$$ where $D_\mu=\partial_\mu+ieQA_\mu$ is the gauge covariant derivative in QED and $F_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A^a_\mu$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the electromagnetic field strength tensor.
For the nonabelian $SU(N)$ gauge theory, $$[D_\mu, D_\nu]=igT^a G^a_{\mu\nu}$$ where $D_\mu=\partial_\mu+igT^a G^a_\mu, G_{\mu\nu}^a=\partial_\mu G^a_\nu-\partial_\nu G^a_\mu-gf_{abc}G^b_\mu G_\nu^c$ are the corresponding field strength tensors.
Though I can derive these results quite easily, I wonder what is the physical content of these two equations. Here a similar question was asked but the answer involves advanced mathematical concepts and notations such as wedge product, exterior derivatives, holonomy etc. Can someone dumb down the notation and state the significance of these relations? Thanks in advance!