I was reading the introduction to Bott & Tu and I did not understand the following geometric characterization of closed forms that is explained there. Let $M$ be a smooth manifold and $\omega \in \Omega^k(M)$ be a $k$-form of $M$. The authors claim that a form is closed (i.e. $d\omega = 0$ ) if and only if given any two $k$-dimensional submanifolds $\Sigma_1$ and $\Sigma_2$ such that the embeddings of these two submanifolds are homotopic relative to their boundaries, then $$ \int_{\Sigma_1} \omega = \int_{\Sigma_2} \omega $$ and conversely if the above integral equality holds for all such embedded submanifolds then $\omega$ is closed. Why are these claims true?
I suppose this gives a very concrete geometric interpretation of a form being closed. Is there a similar geometric interpretation of exact forms (i.e. with no mention of the exterior derivative)?