The exterior derivative is an intrinsic way of talking about the gradient of a function. If you want to understand the intuitive meaning of the exterior derivative of $f$ you should make sure you understand $\nabla f$ properly. I am a little hesitant to post such an answer 5 years into the discussion but as I did not find any occurrence of the string "gradient" on the page I thought this might be useful.

The exterior derivative is an intrinsic way of talking about the gradient of a function. If you want to understand the intuitive meaning of the exterior derivative of $f$ you should make sure you understand $\nabla f$ properly. I am a little hesitant to post such an answer 5 years into the discussion but as I did not find any occurrence of the string "gradient" on the page I thought this might be useful. In the presence of a metric the relation between them is $\langle \nabla f, V\rangle=df(V)$ for tangent vectors $V$.