Its easy to check that every matrix in $SL_2(\mathbb{R})$ can be wrtten uniquely as $\begin{pmatrix} \lambda & \alpha \\ 0 & \lambda^{-1}\end{pmatrix}\begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix}$ where $\lambda>0$. This is exactly written then as a coset representative of the quotient you wrote down above. You can arrive at the hyperbolic metric by following the definition of the pushforward metric from a left invariant metric on $SL_2(\mathbb{R})$. Notice $(\alpha,\lambda)$ is a point in the upperhalf plane.
The latex misbehaved, those should be $2\times 2$ matrices.