Eigenvectors: the directions a matrix doesn't turn
Most vectors change direction when you multiply them by a matrix. Eigenvectors are the special ones that only get stretched or shrunk, and the eigenvalue is the stretch factor.
So Av = λv literally says "A acts on v like plain multiplication by λ."
To find them, solve det(A - λI) = 0 for λ, then find the null space of A - λI for each λ. Picturing the stretch made the algebra feel like it was actually about something.
Comments
The picture of stretching without turning is what made it click for me too.