Why can the :$\mathbf W^*_{MMSE} $ be written as $E[\mathbf s \mathbf y^*]E[\mathbf y \mathbf y^*]^{-1}$
The mean squared error equalizer is one that minimizes the mean squred error 

$$J(\mathbf{W})=\mathbb{E}\left[\|\mathbf{s}-\mathbf{W}\mathbf{y}\|^2\right]$$

which is the equalizer that minimizes the Euclidean distance between the transmitted signal $\mathbf{s}$ and the equalized received signal $\mathbf{Wy}$. To find the value of $\mathbf{W}$ the minimizes $J(\mathbf{W})$, take the derivative with respect to $\mathbf{W}$ and put it equal to $\mathbf{0}$, which will result in 

$$-2\mathbb{E}\left[\left(\mathbf{s}-\mathbf{Wy}\right)\mathbf{y}^H\right]=\mathbf{0}$$
Solve the above equation with respect to $\mathbf{W}$ you get the MMSE equalizer. 

Note: $(.)^H$ is the conjugate transpose, which is equivalent to $(.)^*$ in the paper.