I second the recommendation to at least flip through <i>Gravitation</i>.  It has an intimidating size, but easygoing manner.  I had a lot of difficulty with Spivak's <i>Calculus on Manifolds</i> (which has essentially no physical intuition outside the Archimedes exercise at the end), but I think I was uncomfortable with the abstract notions of tensor product and dual vector space at the time I was learning from it.  At some point I caught on that df was supposed to eat vector fields and produce functions, and things got a little better.

You might try [Sternberg's Advanced Calculus][1] (available on line), especially chapters 11 and 13.

<b>Edit:</b> I like to think of abstract forms as "things to integrate" and Stokes's theorem as some kind of adjunction between boundaries and the derivative.  This becomes a bit more meaningful when homology and cohomology are introduced.  I don't have much advice for connecting with physical intuition, but I have found it useful to:

 1. Decompose div, grad and curl in terms of d and the metric.
 2. Work through some E&M starting from a 1-form (strictly speaking a U(1)-connection) A on $\mathbb{R}^{1,3}$ (see [Wikipedia][2]).


  [1]: http://www.math.harvard.edu/~shlomo/
  [2]: http://en.wikipedia.org/wiki/Maxwell%2527s_equations#Four-potential