Geometric Interpretation of the Exterior Derivative of a 1-form

Question

I was reading this link where says that the geometric interpretation of the exterior derivative of a 1-form $\varphi$ is “the sum of $\varphi$ on the boundary of the surface defined by its arguments” and I have some doubts about that sentence.

1. $d\varphi$ is antisymmetric so by definition it must vanish over linearly dependent vector fields. So the part of the bolded sentence where says “... the surface defined by its arguments” makes sense: is the infinitesimal surface generated by two linearly independent vector fields. Is this reasoning correct?
2. What is exactly the boundary of an infinitesimal surface?
3. And the most important one: what is the meaning of “the sum of $\varphi$ on the boundary... ”? $\varphi$ is a different linear 1-form at each point of the boundary, but those 1-forms live in different vector spaces so, how are you suposed to sum them?

Any other geometric interpretation for the exterior derivative of a 1-form will be welcome.

Thanks in advance!