I don't understand how integration of a $k$-form on a manifold makes sense. For simplicity let's assume $M$ is a smooth $k$-manifold, $\gamma: [a,b] \rightarrow M$ is a smooth curve on $M$ and that $\omega \in \Omega^1(M)$. Then by definition, $$\int_{\gamma} \omega = \int_{[a,b]} \gamma^* \omega$$ In coordinates $(x^i)$ we can write $\omega = \omega_i dx^i$ so that we then have $$\int_{[a,b]} \gamma^* \omega = \int_a^b (\omega_i \circ \gamma)d\gamma^i = \int_a^b (\omega_i \circ \gamma) \frac{d\gamma^i}{dt}dt$$
From an elementary calculus point of view, when integrating, we take the value of $(\omega_i \circ \gamma) \frac{d\gamma^i}{dt}$ at a point on some subrectangle of some partition, which gives us a real number. We then multiply it by the length of the subrectangle (which is approximated by $dt$), which is also a real number and we add these products up, then let the mesh of the partition tend to $0$.
From a differential forms point of view, $(\omega_i \circ \gamma) \frac{d\gamma^i}{dt}$ is a function that is being multiplied by the $1$-form $dt$. The value of $(\omega_i \circ \gamma) \frac{d\gamma^i}{dt}$ at a point of a subrectangle is a real number, but the value of $dt$, since it is a covector field, is a covector. Thus the value of $(\omega_i \circ \gamma) \frac{d\gamma^i}{dt} dt$ at a point of a subrectangle is a covector as well. How does it make sense to sum up a bunch of covectors to obtain a real number (the value of the integral)?
A differential n-form measures oriented n-volumes of supplied n vectors. Those measurements are real numbers and can be summed up.
Integration operator feeds $dx$ with (infinitesimal) tangent vectors at (continuum) consecutive points along a curve: $\int_C dx=\lim\sum dx_{p_i}(v_i)$.