To integrate we need a measure. A measure is a set function, $\mu$, which takes sets as arguments and spits out elements of $\mathbb{R}^*$ (positive real number with infinity included as a point).
We write $\int f d\mu$ for the integral of a function $f$ -approximated by step functions- to mean $\sum \alpha_i \mu(A_i)$; where $\alpha_i$ are the values the step function assumes, and $A_i=f^{-1}\{(\alpha_i)\}$ the set of all $x$ that get mapped to that fixed value.
How do we reconcile this picture with the fact that we can integrate differential forms, $\int A_\mu dx^\mu$. The coordinate function $x^\mu$ is obviously not a set function, hence not a measure. What is even the meaning of $\int A_\mu dx^\mu$? Do we break up $A_\mu$ into a sequence of step functions? How do we define the measure?
For a subset $V$ of $\mathbb{R}^n$, every $n$-form can be written written $\omega=fdx^1\wedge\dots\wedge dx^n$, and you define $$\int_V\omega=\int_Vf dx^1\wedge\dots\wedge dx^n:=\int_Vf dx^1\dots dx^n,$$ where $dx^1\dots dx^n$ is the Lebesgue measure on $\mathbb{R}^n$. On an (oriented) manifold (think about $\int_a^bfdt=-\int_b^afdt$), you define the integral by taking a partition of unity $(\psi_i)$ suboordinate to an (oriented) atlas $\{(\varphi_i,U_i)\}$ and set $$\int_M\omega=\sum\psi_i\int_{\varphi(U)}(\varphi_i^{-1})^*\omega,$$ where the integrals under open subsets of $\mathbb{R}^n$ under the sign $\sum$ are defined as previously. So we patch coherently together Lebesgue integrals in order to define manifolds integrals, and with this formula you can still check some properties from measure theory (example with the change of variables: if $F:M\to N$ is an orientation preserving diffeomorphism, then $\int_MF^*\omega=\int_N\omega$).