Why the integration of a chain representing the fundamental class of a closed oriented hyperbolic manifold is the volume?

Question

Reading a book I've encountered this affirmation:

Let $M$ be a closed oriented hyperbolic manifold, $[M]$ the foundamental class generating $H_n(M,\mathbb{Z})$, $\alpha$ a n-chain in $C_n(M)$ representing $[M]$ and $\omega$ a volume form positively oriented.

We have that $\int_M\omega=\int_\alpha\omega$

There isn't any further explanation, but I have no idea of why this is true.