A number of books not mentioned but particularly useful for the relation to physics:
- Bamberg & Sternberg "A Course in Mathematics for Students of Physics 1&2"
Volume 2 all about physical examples and explanations. Some very neat introductory examples on a basic EM level such as doing linear electric network theory. What's quite exceptional here is how strongly the topological flavor is worked out and many of the early examples are on complexes, hence giving some very solid intuition behind the discrete and continuous cases and their relationship from a topological point of view. Probably the most topological of introductory texts that I know that deal with physics formulated via differential forms.
- Burke's unpublished Div Grad and Curl are Dead
Lots of physical settings here but has banished div/grad/curl, so it's a way to get direct physical intuition from differential forms without any emphasis on relation to old-school vector calculus.
Similar to the mentioned book by Arnold (in that it goes towards Hamilton/Lagrangian mechanics using differential forms hence arriving at symplectic geometry):
- Abraham & Marsden "Foundations Of Mechanics"
Note that relationships of the generalized Stokes theorem to old school Green/Gauss/Stokes theorems are left as exercises. One can find these derivations worked out in Arfken, Weber, Harris or Abraham, Marsden & Ratiu.
- Darling's Differential forms and connections
Provides direct comparisons of old and new Stoke-type theorems. Good for physical examples from Gauge Field Theory.
For the underlying alternating constructions, mentioned by Qiaochu, are actually not from the differential aspect of differential form per se but spring from the exterior algebra, hence can be studied in vector spaces. The original source for these ideas is Grassmann. I recommend his second book, translated in 2000 as a decent point to get some ideas about this. A more modern way to get there is through multilinear algebra. The computation of higher dimensional linear bounded entities is skew-symmetric, which characterizes this need to keep track of sign (coding orientation). The geometric connection of alternation is the need to do bookkeeping of signs (orientations) as one computes exterior product and other operations.
One could certainly do physics with exterior algebra, Grassmann himself gave some examples. I'm not aware of a good modern text that does physics through exterior algebra outside of differential forms, though it would be very instructive to have it.