Integration is about chopping a region into small pieces, then adding up some small quantity on each piece. Thus, integration is about adding up differentials, and the basic integration operation is \begin{equation} f = \int df \end{equation}

In single-variable calculus, such integrals take the form \begin{equation} W = \int F\,dx \end{equation} which might represent the work done by the force $F$ when moving an object in the $x$-direction. The integrand in this case is $F\,dx$, where $F$ is a function of $x$. Similarly, a typical double integral takes the form $\int F\,dx\,dy$, where now $F$ is a function of $x$ and $y$; the integrand in this case is $F\,dx\,dy$.

Thus, differentials, and products of differentials, are the things one integrates!

Differential forms are just integrands, together with rules for manipulating them, using both algebra and calculus.