In addition to Carlo Beenakker's answer that gives Hall via Sperner directly, I think you can also get it by applying Hall's Theorem for Hypergraphs as follows. Let $G$ be a bipartite graph with partite sets $X, Y$, and write $V(X) = \{x_1, \ldots, x_n\}$. For each $i$, define a $1$-uniform hypergraph $H_i$ with vertex set $N(x_i)$ and edge set $\{ \{y\} : y \in N(x_i) \}$.
Letting $\mathcal{A} = \{H_1, \ldots, H_n\}$, we see that $\mathcal{A}$ has a disjoint set of representatives if and only if $G$ has a perfect matching. Now the condition of Aharoni and Haxell's Theorem 1.1 applied to the family $\mathcal{A}$ is clearly equivalent to Hall's Condition on $G$.