Why half of the protein is saturated when [L] =Kd
The Hill equation in that link uses the simplifying assumption that all ligands bind simultaneously and that the binding sites are identical. The reaction is then P + nL = PL, 
and Kd = ( [P][L]^n )/[PL]
Then by rearranging you get the expression in the link.

The equation "fraction occupied binding sites = [L] / [L] + Kd" Is not the Hill equation. It is called the binding isotherm. You can see that if Kd = [L] then the fraction of occupied binding sites (let's call it theta) = 1/2. Thus half of the binding sites are occupied.