Gradient of Total Variation (TV) Norm in Total Variation Denoising
I am by no means an expert on total variation, however I think you should check out this Wikipedia page. It doesn't directly answer your question, but I believe the lemma below illustrates the relationship between total variation and divergence.

There, it gives a lemma that follows from the Gauss-Ostrogradsky theorem and provides a proof for it,

$\int_{\Omega}f\,div(\phi) = -\int_{\Omega} \nabla f \cdot\phi$.

I think for some intution about divergence, it might be helpful to read Wikipedia's explanation.