I went through Linear Algebra I & II where eigenvalues & eigenvectors played a central role many a time and I was left with no elucidation regarding their practical applications.

Now I know that eigen*s (at risk of egregious oversimplification) can characterize systems and transformations, they are fundamentally widely applicable. Think of stretching narrow an arbitrary 2D image on a non-cardinal axis; an eigenvector can be a key factor that describes that transformation.