A good introduction to Maxwell's Equations
Visualising the Div operator
Visualising the Curl operator
The Grad operator (on a scalar field), applied at a point on the surface, delivers the tangent vector pointing up the steepest slope, with its length measuring the steepness. That's pretty easy to visualise.
Richard Feynman commented somewhere that in the distant centuries yet-to-be, when people look back to the nineteenth century, they will remember it for just one thing: it was the century when Maxwell's equations were formulated.