Consider a cusp form $f \in S_2(\Gamma)$ for a certain congruence subgroup $\Gamma$. I would like to understand why $f(z)dz$ is a holomorphic 1-form on the curve $X_\Gamma$. Its invariance is exactly the modularity condition, it remains to understand what holomorphic means and how to prove it.
I don't see at all what to write here even for points away from the cusps (but apparently the difficulty comes at the cusp because of ``ramification").
It's exactly an issue of the local coordinates near/at a cusp: without loss of generality, at $i\infty$ we'd use coordinate $w:z\to e^{2\pi iz/N}$, where $N$ is the width of the cusp. This maps a punctured neighborhood of the cusp to a punctured neighborhood of $w=0$. Then (locally) $z={N\over 2\pi i}\log w$, and $dz={N\over 2\pi i} {dw\over w}$. For this to be holomorphic at $w=0$, we need a further factor of $w=e^{2\pi iz/N}$.