As someone who understands eigenfunctions already, I don't understand the pictures either. Here is the best way to think about it: a matrix is a transformation, a composition of rotation, scaling, etc. Eigensets are lines going through the origin that the matrix moves points along. So a rotation would have no eigenvectors because none of the points move in a straight line, while a scaling along the x axis would have an eigenset that was also along the x axis, consisting of the points that were moved straight up or down.
To imagine finding the eigenset, just ask, could I draw a line through 0,0 such that any point I put on it would stay on it after the matrix acted?
To imagine finding the eigenset, just ask, could I draw a line through 0,0 such that any point I put on it would stay on it after the matrix acted?