Let $\mathbb H \subset \mathbb C$ be the upper half plane. On $\mathbb H$ the group $Sl_2(\mathbb Z)$ acts by Möbius transformations $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} . z = \frac{az + b}{cz + d}.$$ I know that the quotient $Sl_2(\mathbb Z) \backslash \mathbb H$ parametrises elliptic curves, because two elements $\tau, \tau' \in \mathbb H$ define the same lattice $\mathbb Z + \tau \mathbb Z$ if and only if they are in the same $Sl_2(\mathbb Z)$-orbit.
After studying the quotient $Sl_2(\mathbb Z) \backslash \mathbb H$, a common theme seems to study quotients $\Gamma \backslash \mathbb H$, where $\Gamma \subset Sl_2(\mathbb Z)$ is a group of finite index. In particular congruence subgroups $$\Gamma(k) = \{\, A \in Sl_2(\mathbb Z) : A \equiv I_2 \mod k \,\}$$ seem to be of interest. I'm missing some motivation to do that. Why are the $\Gamma(k)$ interesting groups, and why is the quotient $\Gamma(k) \backslash \mathbb H$ interesting?
From the point of view of moduli spaces, considering congruence subgroups of $SL_2(\mathbb{Z})$ lets you consider torsion structure on your elliptic curves.
For instance, $\Gamma(N) \backslash \mathbb{H}$ parametrizes triples $(E,P,Q)$, where $E$ is a complex elliptic curve, and $(P,Q)$ is a basis of its $N$-torsion such that $\langle P,\,Q\rangle_E=e^{2i\pi/N}$ (I think; maybe it’s $e^{-2i\pi/N}$ instead), by
$$\tau \longmapsto \left(\frac{\mathbb{C}}{\mathbb{Z}\tau\oplus\mathbb{Z}},\frac{\tau}{N},\frac{1}{N}\right).$$
You could also wonder about orders of torsion points on elliptic curves, which leads to considering the moduli space of pairs $(E,P)$ where $E$ is a complex elliptic curve and $P$ is a point of exact order $N$ – well, this is exactly $\Gamma_1(N) \backslash \mathbb{H}$ (where $\Gamma_1(N)$ is the subgroup of $M \in SL_2(\mathbb{Z})$ such that $M \equiv \begin{pmatrix}1 &\ast\\0&1\end{pmatrix}\pmod{N}$).
From a modularity point of view, let $L(E,s)=\sum_{n \geq 1}{a_nn^{-s}}$ be the $L$-series of some elliptic curve. We expect that $\Lambda(E,s)=(2\pi)^{-s}\Gamma(s)L(E,s)$ has an entire continuation with $\Lambda(E,s)=\pm \Lambda(E,2-s)$. How could one prove it?
Well, let $f(\tau)=\sum_{n \geq 1}{a_ne^{2i\pi n\tau}}$ for $\tau \in \mathbb{H}$. You can then check that $\Lambda(E,s)=\int_0^{\infty}{t^{s-1}f(it/\sqrt{N})dt}$ (for $Re(s)>>0$).
Our functional equation would then hold if we had $f(it/\sqrt{N})=\pm f(i/(t\sqrt{N}))$. This translates (because $f$ is holomorphic) to $f\mid_2 W_N=\pm f$.
(where $f\mid_k \begin{pmatrix}a &b\\c &d\end{pmatrix}(\tau)=\frac{(ad-bc)^{k/2}}{(c\tau+d)^k}f\left(\frac{a\tau+b}{c\tau+d}\right)$ if $a,b,c,d \in \mathbb{R}$, $\tau \in \mathbb{H}$ and $ad-bc>0$; and $W_N=\begin{pmatrix}0 & -1\\N&0\end{pmatrix}$).
Now it’s clear that $f \mid_2\begin{pmatrix} 1&1\\0&1\end{pmatrix}=f$, so that – as $W_N\begin{pmatrix}1&1\\0&1\end{pmatrix}W_N=-N\begin{pmatrix}1 & 0\\ -N &1\end{pmatrix}$, $f\mid_2 \begin{pmatrix}1&0\\-N&1\end{pmatrix}=f$.
So $f$ would then stable under a large subgroup of $SL_2(\mathbb{Z})$, and it is reasonable to assume that this subgroup is all of $\Gamma_1(N)$.
Actually proving it is, of course, immensely more difficult, and required a huge amount of work with many new ideas.
From a somewhat different point of view, congruence subgroups are exactly those finite subgroup indices of $SL_2(\mathbb{Z})$ on which modular functions with $q$-expansion contained in a number field have bounded denominators.
This was a long-standing problem that was solved recently, see https://arxiv.org/pdf/2109.09040.pdf.