I have been trying to get an intuitive grasp on exterior calculus and found this article particularly interesting :
I can understand why the exterior derivative of a function is visualized using the "level curves", since the derivative is supposed to show the growth of the function and the level curves are more tightly packed where the function grows faster. I can also understand why the wedge product is the intersection of the two sets of level curves, since it's pretty easy to see for the basis $dx^dy$ dx \land dy$ and you can "expand" every 1-form in terms of dx and dy. (I find this one interesting since the wedge does look a little like an intersection symbol...)
Now my question comes with the "d" operator. In the link I gave the author says that it should be visualized as the "boundary" of the set of level curves, which makes sense when you try to interpret stoke's theorem, and also gives an intuitive sense to $d^2=0$, but I can't find an explanation as to why this would be a good visualization?