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.
Yeah rotation matrix are not hermitian (note that you don't need to show the eigenvalues aren't real, you just need to show it's not self-conjugate). The OP may have been confused by the fact that you can make a rotation matrix of eigenvectors of a hermitian matrix, which diagonalises the original matrix into two conjugate rotation matrices with a diagonal matrix between them.
D'oh you are correct! It's a bit late here. Anyways the rest of my point stands about the author's intent. To connect that to your original comment: a rotation is not "observable", instead, it's a transformation that modifies a physical system. In quantum mechanics, physical observables (energy, momentum, position, mass, etc) are always associated with Hermitian operators that act on the Hilbert space of states for the system.
Edit: electricslpnsld correctly points out that the first sentence is false, see below.