Say we have a linear transformation A and some nonzero vector v, and suppose that Av = λv for some scalar λ. This is a very special situation; we say that λ is an eigenvalue of A corresponding to the eigenvector v.
How can we find eigenvalues? Here's one criterion. If Av = λv for some unknown λ, we at least know that Av – λv equals the zero vector, which implies that the linear transformation (A – λI) maps v to zero. If (A – λI) maps v to zero, then it must have a nontrivial kernel, which is to say that it can't be invertible, and this happens exactly when its determinant is zero, because the determinant measures how the linear transformation distorts (signed) areas (volumes, 4-hypervolumes, &c.), so if the determinant is zero, it means you've lost a dimension; the space has been smashed infinitely thin. But det(A – λI) is a polynomial in λ, and so the roots of that polynomial are exactly the eigenvalues of A.