- The statement
I know if we remove the elliptic points then the $\pi_1$ is exactly $\Gamma$.
is wrong. You can see this by considering $\Gamma=PSL(2,\mathbb Z)$.
What you probably meant to say is that if $\Gamma$ is torsion-free, then $\pi_1(H^2/\Gamma)\cong\Gamma$.
- It is a very general theorem due to Armstrong: Let $X$ be a
path connected, simply connected, locally compact metrizable space
and $\Gamma\times X\to X$ a proper discrete group action. Then
$\pi_1(X/\Gamma)$ is isomorphic to the quotient of $\Gamma$ by its normal subgroup normally generated by elements which have fixed points in $X$. (In your case, these are elliptic elements of $\Gamma$.) From this, you get $\pi_1(H^2/\Gamma)$, where $\Gamma$ is any Fuchsian group.
Armstrong, M. A., The fundamental group of the orbit space of a discontinuous group, Proc. Camb. Philos. Soc. 64, 299-301 (1968). ZBL0159.53002.