Here's a short topological proof that $F_2$ contains the free group on countably many generators. The key is that
the classifying space $S^1 \vee S^1$ of $F_2$ has a covering space which is homotopic to a wedge of countably many circles
and this space has fundamental group free on countably many generators by Seifert-van Kampen. The relevant covering space consists of a circle attached to every integer point on $\mathbb{R}$, where the covering map sends the edges between consecutive integers to one loop $y$ in $S^1 \vee S^1$ and sends the circles to the other loop $x$. The fundamental group of this covering space injects into $F_2$, and in fact it is freely generated by elements of the form $y^{-n} x y^n$, as can be seen from the contraction which takes all of $\mathbb{R}$ to a point.