This paper provides an intuition of $k$-forms. On page $4$ it reads:
"In order to represent the exterior product of two $n$-forms you find the intersection of the diagrams representing the forms. For example if we draw $dx$ as vertical lines and $dy$ as horizontal lines then $dx ∧ dy$ is the set of intersection points which will form a uniform grid."
Why can the exterior product of two forms be interepreted as the intersection of the diagrams representing the forms?
This is a follow-up to this post.