I don't get why $\displaystyle \vec p_0=d\vec e$
Because e is a unit vector with unit magnitude. And since the distance to the plane (the length of Po) is d, this means the magnitude of Po is d but the magnitude of e is 1 because its a unit vector. and since e is in the same direction as Po then Po is a constant multiple of e, so its like stretching e. So we must have Po=de
or you could solve the equation Po=xe and find x as follows;
|Po|=|xe|
d=x|e|
d=x*(1)=x