Note: I wrote this when the title was still "What is the divergence? What is the curl?"
A nice geometric interpretation of the divergence is that it measures the rate of expansion of a fixed volume in the flow defined by the vector field. There is a very concrete way to see this by comparing the volume of an small cube to the volume of a parallelepiped given by considering where the corners of the cube are dragged by the flow in an infinitesimal length of time (it is easiest to work out the analogous case of a square with corners (x,y), (x+dx,y), (x,y+dy), (x+dx,y+dy) in the plane). I imagine the fact that the determinant measures volume can be used to explain the presence of the cross product. A more sophisticated (and conceptual) way to prove this fact is to note that the divergence can be defined on any manifold with a volume form as the Lie derivative of the volume form (contract the volume form with the vector field and then take the exterior derivative).
I have also seen a result identifying generalizing the curl with infinitesimal rotations or elements of the Lie algebra of SO(3) to arbitrary dimension by noting that we can identify 2-forms with skew-symmetric matrices (elements of the Lie algebra of SO(n)) if we have an inner product. I'd be curious to hear how exponentiating this infinitesimal rotation relates to integrating the vector field.