I've found them easiest to understand in terms of fluid flow. Imagine that you've got a vector field corresponding to the velocity of a fluid. Div gives you the amount that the fluid expands by -- so for most liquids you'd get zero divergence. Curl is more difficult. Imagine tracing out a close loop, and summing how much net fluid flow there is along the loop -- for the case of an infinitesimal loop, that corresponds to the curl.
i am having trouble with understanding the physical significance of these two operators.
they're somewhat like they sound. Divergence represents change as you get away from a point. Central forces like gravity are divergent: as you get farther away from a source point, the effects change, no matter what direction you go, as long as it's away, in the r-hat direction. (radially outward/inward)
Curl is more synonymous with the magnetic field or turbulent flow. It describes more how things change as you circumnavigate the point. (circumferentially around)
to measure divergence at a point, put a little sphere centered at the point ans measure the total flow of your fluid across the surface of the sphere in one unit of time (i.e. dot the velocity vector of the flow with the normal vector of the sphere and integrate), and divide by the volume of the sphere. then let the radius of the sphere go to zero.
to measure curl, in the plane, put a little circle centered at the point, and dot the vector field with the tangential vector of the circle, i.e. measure the tendency of the field to rotate around the circle, and integrate, and divide by the area of the circle, then let the radius go to zero.
this seems intuitively pretty much like it anyway.