I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have already passed my university courses that have covered these topics, including Stokes's theorem with proof (My uni has this approach of definition - theorem - proof structure in all courses with very little motivation). The issue is that sadly I've never intuitively seen that it could have anything to do with the real area if the manifold would be an earth-like object or any other intuitive geometric structure.
Is there any free material/youtube class that illustrates that it truly corresponds well? I don't need to see proof as I quite understand the technical side of things. My issue also is that I cannot see differential form in any other way than only technical mathematical extremely abstract definition.
Please keep in mind that my knowledge is limited to European standards of a bachelor's degree...