This from wikipedia is where it started to click for me:
"In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix".
So the eigenvectors are like the directions that describe position of an airplane: roll pitch and yaw.
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This from wikipedia is where it started to click for me:
"In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix".
So the eigenvectors are like the directions that describe position of an airplane: roll pitch and yaw.
This point of view was used in a recent HN post: https://news.ycombinator.com/item?id=8904089 .