Can someone provide an explanation of the intuition behind De Rham Cohomology, and what exactly one is trying to achieve?

Assuming the audience knows of manifolds, tangent spaces, cotagent spaces, and differential forms, what is a proper and insightful reasoning to give of the set-up of De Rham Cohomology and how it came to be?

In essence, what is the elevator pitch? And if you have a moment to expand on said pitch, that would also be quite appreciated.

I know, for example, that there's some game going on with exact and closed forms that relates to the existence of holes in a manifold, but would be quite happy to hear a further explanation.