Just think of it as energy.
The higher the mass the more energy you have to get rid of. Ep=mgh
At max limiting IAS you have a fixed max drag which won't change with weight.
So per min you will be able to get rid of xxx amount of energy call it Ed
If you look at Ed=mg(delta)h, g is fixed so the only thing that can vary is the (delta)h which is your rate of decent.
(delta)h = Ed/mg so if you increase m (weight of the aircraft) you decrease (delta)h.
Basic physics