The set $\Gamma\cap\Omega$ is a different set of curves than $\Gamma$. One curve in $\Gamma$ could become many curves in $\Gamma\cap\Omega$. This is how we find $\gamma$.
More specifically, we have a point $x$ with a neighborhood $U$ (guaranteed by discreteness of $\Gamma$ in $S$) that meets only finitely many curves in $\Gamma$, but (by assumption toward contradiction) meets infinitely many curves in $\Gamma\cap\Omega$. Since each curve in $\Gamma\cap\Omega$ comes from some curve in $\Gamma$, there must be a curve in $\Gamma$ which induces infinitely many curves in $\Gamma\cap\Omega$. This curve is $\gamma$.