Every collection of pairwise disjoint non-empty open intervals of $\Bbb R$ is countable. If $\mathscr{I}$ is such a family of intervals, each $I\in\mathscr{I}$ contains some rational number $r(I)$. If $I,J\in\mathscr{I}$, and $I\ne J$, then $I\cap J=\varnothing$, so $r(I)\ne r(J)$. Thus, the map $r:\mathscr{I}\to\Bbb Q:I\mapsto r(I)$ is injective, and it follows that $|\mathscr{I}|\le|\Bbb Q|=\omega$, i.e., that $\mathscr{I}$ is countable. This is the easier part of the theorem.
To finish proving the theorem you must show that every open $S\subseteq\Bbb R$ is a union of pairwise disjoint non-empty open intervals. The easiest way to do this is to define an equivalence relation $\sim$ on $S$ as follows: if $x,y\in S$, then $x\sim y$ iff either $x\le y$ and $[x,y]\subseteq S$, or $y\le x$ and $[y,x]\subseteq S$. In other words, $x\sim y$ iff the entire closed interval between $x$ and $y$ is contained in $S$. To finish the proof you must do two things:
- Prove that $\sim$ actually is an equivalence relation on $S$.
- Prove that each $\sim$-equivalence class is an open interval in $\Bbb R$.
I’ll let you try; if you get stuck, I can add to the answer.