I am self-studying differential forms, and I'd like to understand the proof of Stoke's theorem.

I have read through the document here by Dan Piponi and really liked his explanation of the exterior derivative as finding the boundary in the picture (section 4, page 6), but I have a hard time to connect this with the formal definition of the exterior derivative.

Reading through the comments of his answer here, I am not sure if the concept of 'finding the boundary in the picture' applies to just `xdy`

or any differential forms?