First of all, I know that infinitesimals are not well defined in standard analysis and they have rigorously nothing to do with differential forms. My doubt is on the intuition between one relationship that seems to exist: it seems that differential forms turn the idea of infinitesimal widely used by Physicists into rigorous stuff.
Why I say that? Well, I'll expose some of the points I have noticed.
The total differential. In classical language, given $f : \mathbb{R}^n\to \mathbb{R}$ we can consider the infinitesimal change $df$ when we move from a point $(x^1,\dots,x^n)$ to a neighbouring point $(x^1+dx^1,\dots,x^n+dx^n)$ as being $$df = \sum_{i} \dfrac{\partial f}{\partial x^i}(x) dx^i$$ in differential forms we have $df$ the exterior derivative that gives changes in $f$ when we give vectors to it. So $df$ and $dx^i$ became "measuring objects" in some sense.
Integration along curves. In classical language, if a curve is given parametrized with $x = x(t)$, $y=y(t)$ and $z=z(t)$, then we compute the integral of a vector field $F$ as follows: we consider $dl = (dx,dy,dz)$ an infinitesimal displacement and do the calculation $$\int_\Gamma F\cdot dl = \int_\Gamma F^1dx+F^2dy+F^3 dz = \int_a^b F^1 x'(t)dt+F^2y'(t)dt+F^3z'(t)dt$$ This obviously is related to the pullback. Considering $\gamma$ the parametrization and $F$ a one-form, then $$\gamma^\ast F = \gamma^\ast\left(\sum F_i dx^i\right)= \sum_i F_i\circ \gamma d(x^i\circ \gamma)=(\sum_i F_i\circ \gamma )(\gamma^i)'dt$$ and the integral is the same as before if we define $\int_\gamma F = \int_a^b \gamma^\ast F$
A differential form picks one $k$ alternating multilinear map in each tangent space. In that case, we know we can represent those objects by two $(n-k)$ planes in the tangent space. Intuitively, we can pick just small pieces of those planes, and so at a point, a differential form could be thought intuitively as an infinitesimal $(n-k)$ piece of plane.
Considering all of that it seems that differential forms really exists to make stuff with infinitesimals rigorous. But what's the intuition between this conection between differential forms and infinitesimals?