Rotation matrices are hermitian. It's a popular science article, but the author is alluding to the fact that Hermitian matrices have real eignenvalues. In quantum mechanics and quantum field theory, all _physically observable quantities_ of a system can be expressed as the eigenvalues of a set of a (matrix) operator associated with that quantity acting on basis states in the Hilbert space of the system. This makes sense a posteriori since we have no idea how we would interpret e.g. complex-valued energy or momentum.

Edit: electricslpnsld correctly points out that the first sentence is false, see below.