Physical and Geometrical interpretation of Differential Forms

As it is, differential forms don't tell you the whole story--strictly speaking, differential forms only deals with covectors and wedge products of covectors and then uses the hammer of the Hodge star to be able to clumsily do inner products. To me, it is too far removed from the vector calculus you may already know.

Instead, I strongly urge you to look into geometric algebra. All of the results of differential forms apply to geometric algebra as well--the former is strictly contained in the latter--but the notation is much more familiar and the emphasis is on geometric interpretation instead of abstract symbol pushing. David Hestenes has several books on the subject. Probably the authoritative piece as far as using geometric algebra to solve physical problems is Geometric Algebra for Physicists by Doran and Lasenby. You can also read some things quickly in this website, written by Gull, Doran, and Lasenby.

I'll give a quick overview. Geometric algebra has a wedge product like differential forms does, but it also lets you directly use a dot product as well. In fact, it combines the two in a useful operation called the geometric product, defined as follows. For two vectors $a, b$, the geometric product $ab$ is

$$ab = a \cdot b + a \wedge b$$

The geometric product is associative (even though the dot product is not!). This makes it very useful. It is also invertible in Euclidean space, as a consequence of that associativity. This makes possible the formula

$$a = abb^{-1} = (a \cdot b) b^{-1} + (a \wedge b) \cdot b^{-1}$$

Geometrically, this decomposes $a$ into $a_{\parallel, b}$ and $a_{\perp, b}$. We emphasize that $a \wedge b$ denotes an oriented plane, and further wedge products yield oriented volumes and more.

Some applications immediate to physics are as follows:

1. Angular momentum as a bivector. This is one of the first times you "need" a cross product, and using the wedge product instead yields a cleaner interpretation. The angular momentum bivector is exactly the plane in which two objects move in relation to one another. This also generalizes to beyond 3d, so it makes sense to talk about angular momentum bivectors in relativity also.
2. Unification of integral theorems (the fundamental theorem of calculus). Geometric calculus (like differential forms) makes possible the unification of the divergence theorem, Stokes' theorem, and so on as one basic concept: that the integral of a function over a boundary is equal to the integral of the derivative over the region bounded by that boundary. I think this is a significant quality of life issue; having to remember only one concept is much easier, in my opinion, than remembering many separate integral theorems.
3. Relativity without indices or classic tensor calculus. Geometric algebra's combination of the dot and wedge products makes possible all the usual operations one usually needs tensor calculus and index notation for. Relativity can be presented using a modest extension of the methods used in 3d electromagnetism. The geometric product makes it possible to boil down Maxwell's equation in vacuum to one equation (instead of two for differential forms): $\nabla F = J$. This emphasizes the interpretation of the EM field $F$ as a bivector field, a field of oriented planes throughout spacetime.
4. Geometric interpretation of quantum mechanics. A lot of the mathematics in quantum is presented as mystical or special to QM, but most of it is actually inherent to the geometric structure of space and time. Geometric algebra allows one to treat the Pauli and Dirac algebras as the algebras of basis vectors in 3d and 3+1d space. This makes the interpretation of spin and spin operators inherently geometrical.
5. Construction of spinors. Spinors are things we often deal with in quantum, perhaps with only the understanding that they must be rotated through $4\pi$ instead of $2\pi$ to return to their original form. Geometric algebra shows that spinors underlie all rotations--even those of plain old 3d space. In fact, the spinors of 3d space are quaternions, and the spinors of 2d space are complex numbers. GA gives a framework for constructing spinors and manipulating them like other objects.

Differential forms can do some of these things as well, some of them not (it absolutely can't boil down Maxwell's equations to one expression). Either formalism is a great improvement over traditional methods, however.

I would really recommend the book by Frankel, The Geometry of Physics. He deals with all the fundamental concepts of topology and differential geometry, but gives clear and detailed applications to classical mechanics, electromagnetism, GR and QM. He is not too formal, but develops really a lot of useful tools using differential forms.

Another book, which is a bit more basic and is so to say a light-version of the classical Arnold's Mechanics textbook is Geometric Mechanics, by Richard Talman. One can develop a geometrical and a physical intuition of differential forms. Here the applications are mostly reduced to classical mechanics.

There are of course other good texts, but these are really good starting points.

I also agree with @Muphrid, that Geometric Algebra should be actually preferred as the language for modern physics, instead of differential forms. It is much clearer and familiar. Check the book by Lasenby and Doran and also the dissertation of Anthony Lewis at cosmologist.info, who has a chapter dealing only with the translation between differential forms and geometric algebra.

This may not be exactly what you are looking for, but I am going to recommend two specific texts.

Misner, Thorne, and Wheeler, Gravitation, Chapters 4, 9, and end of 14

Solidly in the realm of physics but they have a lot of tidbits of interpretation in there.

Choquet-Bruhat and DeWitt-Morette, Analysis, Manifolds, and Physics, Chapter IV.C

I mean, this text is incredible. We should all read all of it, all the time. More mathy but that's better in this case. Also more applicable then just E/M + Gravity like MTW. A bit old, but it's nice notation for breaking you in.