at.algebraic topology - How does this geometric description of the structure of PSL(2, Z) actually work?

The key word here is "Bass-Serre Theory" -- using the action on the hyperbolic plane, you can easily cook up a nice action of $PSL_2(mathbb{Z})$ on a tree. This is all described nicely in Serre's book "Trees".

EDIT: Let me give a few more details. It turns out that a group $G$ splits as a free produce of two subgroups $G_1$ and $G_2$ if and only if $G$ acts on a tree $T$ (nicely, meaning that it doesn't flip any edges) with quotient a single edge $e$ (not a loop) such that the following holds. Let $e'$ be a lift of $e$ to $T$ and let $x$ and $y$ be the vertices of $e'$. Then the stabilizers of $x$ and $y$ are $G_1$ and $G_2$ and the stabilizer of $e'$ is trivial.

If you stare at the fundamental domain for the action of $PSL_2(mathbb{Z})$ on the upper half plane, then you will see an appropriate tree staring back at you. There is a picture of this in Serre's book.

EDIT 2: This point of view also explains why finite-index subgroups $Gamma$ of $PSL_2(mathbb{Z})$ tend to be free. If you restrict the action on the tree $T$ to $Gamma$, then unless $Gamma$ contains some conjugate of the order 2 or order 3 elements stabilizing the vertices, then $Gamma$ will act freely. This means that the quotient $T/Gamma$ will have fundamental group $Gamma$. Since $T/Gamma$ is a graph, this implies that $Gamma$ is free.