...why wave functions are represented as eigenvectors?
Because it makes the math easier.
Technically - the wave-function is a state-vector.
We like to resolve them into eigenvectors (also called eigenfunctions) for the same reason we like to resolve a velocity vector into components: it makes the math easier.
All wave-functions are vectors because they transform as vectors - so "function" and "vector" mean the same thing.
Do you even know what an eigenvector is? The reason I ask is that four hours ago you asked "Hey all, does anyone a great place to learn linear algebra online? Thanks for any help. " It took me more than four hours to learn linear algebra.
Yep - after grokking that, it remains only to figure which representation of vectors is the one you want to deal with. Some of the cooler ways can look a lot like magic.
This gets very useful when you have to deal with much more complicated systems where writing down the functions normally can occupy a whole page - it can also obscure the relationships that you are interested in.