I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their properties, defined forms, learned of their pullbacks and the properties of these pullbacks, and defined the differential operator while learning some of its properties. I am currently reading about exact/closed forms in the build up to a certain "Poincare Lemma".

While the theory all seems to be fitting together (albeit with a bit of effort), there has been a nagging question. What is the motivation here? It has been my experience that many mathematical constructions (that I have encountered at least) are done with the goal of better understanding something. I feel like this thing is missing from my understanding of differential forms. Any insight will be appreciated.