A matrix moves every vector in the plane. Most vectors get knocked off their line. An eigenvector is one that stays on its own line: the matrix only stretches it, shrinks it, or flips it.
The stretch factor is the eigenvalue:
A v = λ v
Rewrite it as (A − λI) v = 0. A non-zero v can only be sent to zero if A − λI squashes space flat, and squashing flat is exactly what a zero determinant means:
det(A − λI) = 0
That equation, the characteristic polynomial, is where the eigenvalues come from. Each one then gives its eigenvectors by solving the linear system.
For A = [[2, 1], [1, 2]] the polynomial is (2 − λ)² − 1, so λ = 1 or λ = 3. The line y = x is stretched by 3; the line y = −x is left alone.