If you find that anything besides a poor dry explanation, that is on you. I am sorry this enrages you.

Eigenvalues in an oscillating system describe its resonant frequencies. Its eigenvectors can describe motion at that certain resonant frequency. Imagine a bridge. If wind or traffic match a resonant frequency (eigenvalue) it would be dangerous. Engineers can redesign it to change the corresponding eigenvector and shift the eigenvalue (its resonant frequency for that mode of oscillation) to a safer range. See that bridge in London.