The exterior algebra arose naturally in a problem about enumerating matchings on a planar graph (domino/lozenge tilings). In some situations, those correspond to families of nonintersecting lattice paths between a set of sources $S$ and a set of sinks $T$. The Lindstrom-Gessel-Viennot lemma says that the number of these families equals the determinant of a matrix $M$ whose $i,j$ term is the number of paths between the $i$th element of $S$ and the $j$th element of $T$.
Suppose you want to count the domino tilings of a modified region where only some subset of $S$ is present and only some subset of the same size of $T$ is present. This happens if you have a large region you cut into pieces along $S$ and $T$, and want to enumerate tilings of the larger region by the transfer matrix method. If you index a matrix by subsets of $S$ and subsets of $T$, and put the number of domino tilings of the region in that entry, the result is the action of $M$ on the exterior algebra.