[Tommah]

bee_rider already touched on this in another comment, but the theorem makes sense if you consider a matrix with large diagonal values and small off-diagonal values (in magnitude). If I have a matrix with 1,000,000 on the diagonal and 1 everywhere else, I'd expect the eigenvalues to be 1,000,000 plus or minus some small error. The Gershgorin disk theorem proves this and puts an upper bound on the error.

The diagonal elements of matrices have a lot of rather "magical" properties if you think about it. Their sum is also the sum of the eigenvalues of the matrix. And if you have a matrix A that is singular, you can choose any value x that is not an eigenvalue, and then A - xI is invertible but still mostly behaves like A.