If you feel that pullbacks by charts conceal the intuition, you can work with manifolds that are embedded in R^n. Differential forms are still the "right way" to do integration on these manifolds but now can be defined in terms of the coordinate system of R^n. Then you can do cute tricks like seeing that if you want to know the area of a region on the plane, you can compute it by integrating dx ^ dy over the area. OR you can compute it as x dy over the boundary since d(x dy) = dx ^ dy. (Where ^ is the wedge operator.) This means you can And x dy can be integrated mechanically by a planimeter [1]. And this is also how you would compute the area of a region in software given its boundary!

There's some other uses. They form the basis of De Rham Cohomology [2] which is a useful and computational way of describing topological properties of a manifold (recall how Stokes's theorem and friends show how the topology of a space constrains the integrals of differential forms).

Another thing is that they're specific kinds of tensors on the manifold. Tensors represent basically all the information we might be interested in about a manifold (for example, its curvature). Differential 1-forms are "dual" to vectors and are therefore important to building more complex tensors (higher tensors take in some number of vectors and 1-forms and output a value).

And just as regular integration and differentiation relates to solving of differential equations, differential forms are needed for differential equations that are on a manifold.

[1] https://en.wikipedia.org/wiki/Planimeter [2] https://en.wikipedia.org/wiki/De_Rham_cohomology