In linear algebra, an eigenvector of a linear transformation is a non-zero vector
that only changes by a scalar factor (its eigenvalue) when that linear transformation is
applied to it.

Now for the sake of simplicity lets assume that Energy* as a linear transformation, and when it acts on some position (x1,x2) gives you the energy at that point (e0).

(x1,x2) – Eigenvector, e0 – Eigenvalue.

This e0 that you get is a physical measurable quantity and you do not want this value to be complex. Why ? Complex energies are not a thing of the real world.

And the reason why Hermitian matrices are important in Physics is because if a Matrix is hermitian, then it has real eigenvalues.

Thanks for asking!

* It need not be Energy, it could be any physically measurable quantity. We have just taken energy as an example here.

** A Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose ( A = A ^{†} )