(How) is category theory actually useful in actual physics?

Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at Microsoft's Station Q.

Categories (and higher categories) seem to be a good way of expressing the locality of the path integral in physics. In particular, it is the idea of gluing of local structures that is important. This line of thought leads to the axiomatization of (parts of) various QFTs, with the most success in topological and conformal field theories. This idea has its origins with Atiyah, Segal, Baez-Dolan, Freed and probably a ton of other people I'm forgetting. Braided fusion categories as in the previous answer are an example of this in three dimensions. Most recently, there's Lurie's classification of TQFTs in all dimensions in terms of $(\infty,n)$ categories.

Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.

See J. Fuchs, I. Runkel, C. Schweigert: TFT construction of RCFT correlators I, II, III, IV, V for the full story or, for a summary, Schweigert's 2006 ICM talk Categorification and correlation functions in conformal field theory.