Application 1: Such a construction shows that a finitely-presented group acts freely and cocompactly on a simply connected simplicial 2-complex.
Using minimal tracks, Dunwoody deduced that finitely-presented groups are accessible, in the sense that they may be written as the fundamental group of a finite graph of groups whose edges are finite groups and whose vertices are groups with at most one end. As corollaries, we get Stallings' theorem or the stability of virtually free groups up to quasi-isometries (see Lectures on Geometric Group Theory, theorem 18.38, p. 473).
Application 2: In the article Topology of finite graphs, Stallings show that, if $X \to Y$ is an immersion from a finite graph to a finite bouquet of circles, then it is possible to add edges to $X$ in order to construct a covering $Z \to Y$. Such an argument can be used to prove that finitely generated free groups are LERF.
The construction was generalized by Wise (see for example A combinatorial theorem for special cube complexes) for Salvetti complexes, a kind of higher dimensional bouquet of circles; but Salvetti complexes are just the canonical CW-complexes associated to right-angled Artin groups (or RAAGs) with some additional cells of dimension $\geq 3$. Therefore, in the same way, separability properties may be proved for RAAGs, such as residual finiteness:
Let $\Gamma$ be a RAAG, $g \in \Gamma \backslash \{1\}$ and $S(\Gamma)$ the associated Salvetti complex. The universal covering $\widehat{S}$ of $S(\Gamma)$ is a CAT(0) cube complex (whose 2-skeleton is the Cayley complex of $\Gamma$). Thinking of $g \in \Gamma= \pi_1(S(\Gamma))$ as a loop in $S(\Gamma)$, let $\widehat{g}$ be one of its lifts in $\widehat{S}$, and $Y$ be the intersection of all half-spaces containing $\widehat{g}$. Noticing that $Y \subset \widehat{S}$ is a finite convex subcomplex, we have a local isometry $Y \to S(\Gamma)$. From Wise's construction, we get a finite covering $Z \to S(\Gamma)$ such that $Y$ is naturally a subcomplex of $Z$. Therefore, $\pi_1(Z) \leq \pi_1(S(\Gamma)) \simeq \Gamma$ is a finite-index subgroup not containing $g$.
More generally, the same argument proves that RAAGs are truncated subgroup separable (following the definition 5.2 in Elisabeth Green's thesis).