Why can we know the best precoding direction after we do the SVD to the channel?
First you have to know that the vector in $U$ and the vector in $V$ are orthonomal,that is ,$\vec u \vec u^H=I$,and $\vec v \vec v^H=I$.

So now $H=U\Sigma  V^H$,and $y=Hs+n=(U\Sigma  V^H)s+n$

and $U^H y=\Sigma  V^Hs+n$,see the $\Sigma  V^Hs$,isn't it like " channel gain $\times$ beamforming $\times$ signal $s$ "?

Now you can also find that according to the rule of multiplication of matrix,the first element of $\Sigma$,which is the $\sigma_1$,will multiple every element of $v_1^H$,that is why we say the $v^H_1$ is the best precoding ,because its elements will multiply the biggest singular value ,$\sigma_1$, every time.