I like the physics/engineering approach to Stokes theorem. That yields a little more intuition. If you want a clean proof, then the place to look is differential forms, but that takes a little effort to learn (and if you understand differential forms well enough, you can see how it relates to the physics intuition). A good place to learn about differential forms is Mathematical Methods of Classical mechanics, by Vladimir Arnold--he's the only one I've seen so far who explains them in a way that meshes with the physics/engineering intuition.
The curl of a vector field is like an infinitesimal line integral. That's basically the idea. You want to integrate around a big loop, so you break it up into lots of little tiny loops. In the limit, the little tiny loop integrals approach the curl of the vector field dotted with a normal vector. So you add up the curl near each point to get the integral around the big loop.
You can draw a little picture and calculate what that infinitesimal loop integral is for a tiny square parallel to each coordinate plane. Those are the components of the curl vector field.
It's hard to explain very clearly without being able to draw pictures and stuff, but that's the general idea. A good book on electromagnetism, for example, should explain this in more detail.
