It is known that the full modular group $\Gamma=SL_2(\mathbb{Z})$ has the following congruence subgroups $\Gamma(N),\Gamma_0(N),\Gamma_1(N)$ for some positive integer $N$ defined as $$\Gamma(N)=\bigg\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in\Gamma:\begin{pmatrix} a & b \\ c & d\end{pmatrix}\equiv\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} \ (\text{mod} \ N)\bigg\} \\ \Gamma_1(N)=\bigg\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in\Gamma:\begin{pmatrix} a & b \\ c & d\end{pmatrix}\equiv\begin{pmatrix} 1 & * \\ 0 & 1\end{pmatrix} \ (\text{mod} \ N)\bigg\} \\ \Gamma_0(N)=\bigg\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in\Gamma:\begin{pmatrix} a & b \\ c & d\end{pmatrix}\equiv\begin{pmatrix} * & * \\ 0 & *\end{pmatrix} \ (\text{mod} \ N)\bigg\}$$ So clearly we have $\Gamma(N)\subseteq\Gamma_1(N)\subseteq\Gamma_0(N)\subseteq\Gamma$. Define $\mathcal{H}=\{z=x+iy\in\mathbb{C}:x,y\in\mathbb{R},y>0\}$. Then $GL_2^+(\mathbb{R})$ acts on $\mathcal{H}$ as a group of holomorphic automorphisms by $$\gamma:z\mapsto \frac{az+b}{cz+d} \ \ \text{for} \ \ \gamma=\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in GL_2^+(\mathbb{R})$$ and extending this action to $\mathcal{H}^*=\mathcal{H} \ \cup \ \mathbb{Q} \ \cup \ \{\infty\}$ by $$\begin{pmatrix} a & b \\ c & d\end{pmatrix}.\infty=\frac{a}{c} \ \ (c\neq 0)\\ \begin{pmatrix} a & b \\ 0 & d\end{pmatrix}.\infty=\infty \\ \begin{pmatrix} a & b \\ c & d\end{pmatrix}.\frac{r}{s}=\frac{ar+bs}{cr+ds}$$ for rationals $\displaystyle\frac{r}{s}$ with $\gcd(r,s)=1$. That completes all the prequisites of my question.
Now one of my textbooks says that "If $G$ is a discrete subgroup of $SL_2(\mathbb{R})$, then the orbit space $\mathcal{H}^*/G$ can be given the structure of a compact Riemann surface". I didn't understand why this has to happen. Again they say that if $G=\Gamma(N),\Gamma_0(N),\Gamma_1(N)$, then the respective orbit spaces $\mathcal{H^*}/\Gamma(N),\mathcal{H^*}/\Gamma_0(N),\mathcal{H^*}/\Gamma_1(N)$ are called modular curves. But how these orbit space structures can be visualized as some algebraic modular curves I don't understand also. Any help is appreciated.