Eigen-things can also be thought of "fixed values" of a "thing" transformation.
For example - the eigenfunction of a derivative is e^x since when you run the derivative function on e^x you get.... e^x
For example - the eigenfunction of a derivative is e^x since when you run the derivative function on e^x you get.... e^x
For any vector space V over a field F, the set of functions V -> V is a vector space over F. This is essentially because the vector space properties are carried over "point-wise".
When you consider only certain classes of functions (such as L^2), the question of whether they form a vector space boils down to the question whether those classes are closed under addition and scalar multiplication. All other vector space properties are satisfied because they are in the larger vector space of all functions.
Unfortunately, the history of the eigen*s is rooted in the opaque and cryptic (but rigorous) linear algebra definition, and the non-rigorous but meaningful geometry came later.
Fixed directions. The operator operating on an eigenvector doesn't change the vector's direction, but can change its magnitude (i.e. length). (Except, if you change the length from positive to negative, then you kind of change, or flip, the direction to 180 degrees opposite.)