I am always confused by the notations of integral. Consider a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$. The following two notations $\int_a^bf$ and $\int_a^bf(x)dx$ are the same thing. However, to my understanding, $dx$ is the differential of the identity map. Thus, $dx$ is a linear function from $\mathbb{R}$ to $\mathbb{R}$, i.e., $dx(a)=a$.
Then, $f$ and $f(x)dx$ are two different functions. Why are their integrals the same?
This is because the integral of a differential form was defined to mimic the integral of functions such as in your case. Ie, $$\int_{[a,b]} f(x)dx=\lim \sum f(x_i)dx(a_i)$$ where the $a_i$ are the patches in some partition of the interval $[a,b]$ and $\lim \sum$ is admittedly sloppy notation for a Riemann sum.
The idea is that while in the ordinary integral you input a measure by hand, i.e you partition the interval of integration and add a "$\Delta x_i$" to account for the length of each patch to the definition of integration, in the integration of differential forms you let the differential form itself calculate the length (or area/volume in the higher dimensional cases) of the patches in the partition.
In short, the integrals are the same because the differentials of the identity map provide the same (Lebesgue) measure for the integral of the differential form