There isn't any ``interpretation'' of the statement, since a mathematical statement isn't subject to interpretation. It has consequences-it implies *other* mathematical statements and, insofar as Hermitian operators can be used to describe natural phenomena, the statement has consequences that are subject to experiment.

One such consequence is that the eigenvalues don't have any non-trivial phase, whose consequences, in turn,  can be described in many, different but equivalent, ways. (Of course complex numbers are experimentally observable, as well as real numbers.)  That the Hamiltonian, for example, is Hermitian implies that the evolution operator of the system in question is unitary. If not, then the system can't be considered as isolated and must be suitably completed: the imaginary part of an eigenvalue of the Hamiltonian means that the eigenstate in question isn't stable, so the question then arises, how to describe the degrees of freedom, whose dynamics can complete that of the system and give, thus, rise to a Hamiltonian that is Hermitian. 

Of course it is a mathematical statement, which is in fact proven very easily.  I spare myself in  using  interpretation as a terminology but definitely claim that it has a strong physical consequences due to the fact that one may  only observe experimentally the  images of  those mental events or abstract conceptions , which are  not taking place  in real time and space domain , but they  are expressed in   mathematical forms,  and at the final stage are represented by the real numbers or 'metrics'  similar  to the counter parts of the mental images of    those real events   occurring  in the nature spontaneously.

This doesn't mean that imaginary parts (or the phase ) may not carry any valuable information associated with those events but mostly we don't know how to handle them in practice!!!  BEST REGARDS

In quantum mechanics dissipative part of the energy is never considered properly as it has been done in Newtonian mechanics. The most of the dynamical processes such as the acoustic wave  propagation in solid media,  the imaginary part is  associated with the dissipation if  one treats  the problem by complex algebra.  etc.  Similarly the fourth dimension in generalized space and time domain is represented by an imaginary number  i c t. 

We have the same bad habit to throw out the screw  (anti-self conjugate) part of the second order  deformation tensor or dyadic while dealing with the theory of elasticity. 

Of course it's known how to handle complex numbers in practice-that's not the problem. There  exist transformations of physical systems, that are described by Hermitian operators, which means that the phases of the eigenvalues all vanish-and this statement has physical relevance. There exist other transformations, that are not described by Hermitian operators, which means that the phases of the eigenvalues do not vanish-and this statement, too, has physical relevance (cf. previous message for an example). 

I have had a non-Hermitian relaxation matrix when I was dealing with the kinetics of hopping motion of interstitials with chemical reactions in arbitrary time dependent inhomogeneous interactive fields in body centered cubic crystals.  Non-Hermitian arises due to asymmetry associated with the potential energy barrier between the octahedral and tetrahedral sites in BCC  lattice.   This problem was overcomed by using the similarity transformation (not necessarily orthogonal) to transform the non-Hermitian 6x6  relaxation matrix to a Jordan canonical form.           

Dear Artur, for me  there is nothing straight forward and absolutely convincing  in positive sciences  with an exception of MATH. if your are not shy enough to ask WHY.

Best Regards

Dear Artur, If you have natural talent in mathematics, and you have spent all your life solving mathematical problems, and/or   similarly try to put the physical problems into the mathematical format like playing chest  then  you go few  steps further and try to find  the simplest  and most elegant solution of those problems. I don't think that  de Broglie  and Schrodinger who have  had very sound training in mathematical analysis and good  knowledge in the  classical wave optics as well  as in Hamiltonian mechanics to establish  the mathematical foundations of  quantum wave mechanics without too much effort. For me the most genius approach is due to Heisenberg and Dirac in the matrix formulations of Non-relativistic and relativistic formulations of  quantum mechanics, respectively.

Thanks to the mathematicians, who have had discovered the commutation rules of matrix operators long time before.    Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition. Early matrix theory had limited the use of arrays almost exclusively to determinants and Arthur Cayley's abstract matrix operations were revolutionary. He was instrumental in proposing a matrix concept independent of equation systems. In 1858 Cayley published his Memoir on the theory of matrices in which he proposed and demonstrated the Cayley-Hamilton theorem.

NOTE:The Chinese text The Nine Chapters on the Mathematical Art written in 10th–2nd century BCE is the first example of the use of array methods to solve simultaneous equations. In 1545 Italian mathematician Girolamo Cardano brought the method to Europe when he published Ars Magna.

Dear Artur my Erdös number is infinity since I haven't  collaborated with any one who has  published in one of those Mathematical Journals.  Best Regards