If you are a low-dimensional topologist and you care about skein modules (e.g. those which are related to generalized Jones polynomials), then you should also care about higher categories. To compute the skein module of a manifold that has been cut into several pieces you need to use higher categories in an essential way.
Very similarly (and much less surprisingly), if you are interested in computing low codimension TQFT invariants (vector spaces and linear maps), then you should also care about the high codimension TQFT invariants (n-vector spaces), as these are often the most efficient way to do computations.
