I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since it is not immediately compatible (to me) with the physicist POV. How do the two fit in?
Let's start with a euclidean space for a moment, but impose upon this a general curvilinear coordinate system.
The tangent vectors to the coordinate lines through a given point define the usual basis vectors, which are called various names. They constitute vector fields, at any rate, and for the purposes of this answer, I'll call them only the tangent basis vectors (with fields being implied).
These tangent basis vectors are not necessarily orthogonal, and as a result, one can form hyperplanes from $n-1$ of them and find the normal vectors to those hyperplanes. You can choose a particular normalization of these vectors so that a particular tangent vector and its normal counterpart (defined by the normal to the hyperplane formed by all other tangent basis vectors) have unit inner product. These particular normal vectors are called, variously, the dual basis vectors, or the cotangent basis vectors.
All of the above applies in a setting with a metric--in particular, the normal vector requires a metric to be defined as orthogonal to all those vectors in the hypersurfaces.
The leap forward is to consider the case when you don't have a metric; you can still define linear functionals, or forms, such that the form applied to a particular tangent basis vector yields 1, and these forms still span their own vector space, the dual vector space.
Now I'll stop right there, actually, and not go back to the idea of what happens when you don't have a metric again--because in physics, 99% of the time you do still have a metric, and forms are not necessary. You can get by just fine by using those cotangent basis vectors and their linear combinations and wedge products. They obey the same algebra as forms without being as abstract.
What you should understand is that differential forms made the work of doing calculus on manifolds a lot easier, but it's written for the general setting of a manifold that might not have a metric. This results in notation that is, for many physics applications, overly handcuffed. Often, metrical operations get abstracted out by using Hodge duality, for instance. Why? Because asserting a volume form is a weaker condition than asserting a metric.
Differential forms' typical notations can lead to confusion over what's geometrical and what's not. @littleO said (paraphrasing, with slight tweaks) that a differential form integral can be thought of as chopping up a manifold into $k$-vector pieces and then plugging that $k$-vector into into the $k$-form that is being integrated. Let's take that to its logical end: that means, when you integrate a volume form $dV$, the geometry of the manifold isn't coming from $dV$! It's coming from that $k$-vector that is being plugged in.
Differential forms are commonly used for integration in these settings because the metric does not appear in their integrals. That, and the coordinate free manipulations they enable? That's what makes them useful to a physicist. But it also leads to misunderstandings, like thinking that you can no longer integrate vector fields (you absolutely can).
At any rate, for most applications in physics, you can just think of differential forms as vectors (or higher dimensional things, like planes and volumes and so on), just described using that cotangent basis, instead of your regular old tangent basis.