The exterior derivative business is a way to systematize the sorts of derivatives that are already used in Euclidean 3-D vector calculus: the gradient, the divergence, the curl.
Then some of the amazing facts about vector calculus can then be seen as special cases of much more general facts. For example, we know that the curl of the gradient of a scalar is always zero. Similarly, the divergence of the curl of a vector is always zero. In exterior calculus, these are two instances of the more general fact that ##d^2 = 0##. Applying the exterior derivative twice always produces zero.
Then there are two facts relating integrals of different numbers of dimensions:
##\int \vec{A} \cdot \vec{dl} = \int (\nabla \times \vec{A}) \cdot \vec{dS}##
(integrating a vector field around a closed loop produces the same result as integrating the curl of the vector over the surface enclosed by the loop)
##\int \vec{A} \cdot \vec{dS} = \int (\nabla \cdot \vec{A}) dV##
(integrating a vector field over a closed surface produces the same result as integrating the divergence of the vector over the volume enclosed by the surface)
In the exterior calculus, these are two different instances of the more general "Stokes theorem", which applies to objects of any degree.