It's been some time I had to deal with these details of semiconductor physics. Here's what I think I still understand:
In a block of material no two electrons can have the same state (
Pauli exclusion principle). Therefore for a meterial to be able to accomodate a given number of electrons the material has to provide a set of different states, one for each electron. The total number of possible states is given by N(E).
Not every state is occupied by an electron, however. The ratio of occupied states to total number of states is the occupancy.
To calculate the effective numbe rof electrons in any band you multiply the total number of states (max. possible number of electrons) by the ratio of occupied states to total number of states (F(E)) and integrate this number over the respective band (here: conduction band).