The 2D divergence theorem is to divergence what Green's Theorem is to Curl.
If we wanted to find the outward flow instead of the circulation, much like Green's Theorem split it up into tiny bits of circulation addiing them up, the 2D Divergence theorem splits it up into tiny bits of outward(or inward) flow and then adds them all up.

In essence, the 2D divergence theorem allows to transform the flux integral for what could be a tricky curve into a simpler surface integral, or vice-versa, depending on whatever is the easiest case to solve.

It is typically seen in two equivalent forms,
Recall that the normal vector for two dimensions is defined for a vector function as
With the surface integral being defined