why eigenvalues concerned in Harris corner detection?
Basic linear algebra. M is a symmetric matrix, so there's a decomposition:

$M=Q^{\mathsf{T}} \Lambda  Q$

where Q is a rotation and $\Lambda$ is a diagonal matrix.

We can rotate $[\text{$\Delta $x},\text{$\Delta $y}]$ by $Q^{\mathsf{T}}$

$Q^{\mathsf{T}} [\text{$\Delta $x},\text{$\Delta $y}]^{\mathsf{T}}=\left[\text{$\Delta $x}',\text{$\Delta $y}'\right]^{\mathsf{T}}$

giving 

$S(x,y)=\left[\text{$\Delta $x}',\text{$\Delta $y}'\right]\Lambda \left[\text{$\Delta $x}',\text{$\Delta $y}'\right]^{\mathsf{T}}=\Lambda _{1,1} \left(\text{$\Delta $x}'\right)^2+\Lambda _{2,2} \left(\text{$\Delta $y}'\right)^2$

So $S(x,y)$ is large in both directions $\text{$\Delta $x}'$ and $\text{$\Delta $y}'$, if both eigenvalues are large.