Where does one more '$rm m$' come from in the units?

The del operator is

$$nabla = [frac{d}{dx},frac{d}{dy},frac{d}{dz}] $$

and ends up taking derivatives of A with respect to x,y,z axis, which are displacements measured in meters.

Since a derivative is a rate of change, you are finding the change in A(Wb/m) per meter, hence $$frac{Wb/m}{m}=Wb/m^2$$.

As you suggested the del operator provides the additional “per/meter”. The del is the rate of change with respect to position. So just like taking a derivative with respect to time divides the units by time so also taking a derivative with respect to position divides the units by distance.