None of the posted answers can be read in 30 seconds. So despite being 13 years late, here’s one that does actually try to meet that criterion:

https://qpr.ca/blogs/wp-content/uploads/2025/04/Eigenvectors2.m4a

For any movement of points in a set, an eigenvector for that movement is a line between two points in the set for which the movement does not change its direction. For example in physical 3d space if the movement of points is a pure expansion then every such line is an eigenvector (with eigenvalue being the expansion factor), and if the movement is a rotation then the eigenvectors just point along the axis (and the corresponding eigenvalue is 1).

Source: (194) Alan Cooper’s answer to Can you explain eigenvectors in 30 seconds to a layperson? – Quora