We compute the homology of the matching complex $M(Γ)$, where $Γ$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard techniques in combinatorial topology, such as the theory of shellings. We then broaden the scope to the larger class of fibre-closed families of simplicial complexes and consider these through the lens of representation stability. This allows us to prove a number of results of an asymptotic nature, such as an analysis of the growth of Betti numbers and the kinds of irreducible $S_n$-representations that appear.