The $sp^2$ bonds can be regarded as approximately localised because each bond only involves two carbon atoms. I suppose they will form bands in graphene, but very narrow ones. Anyhow, when two $sp^2$ orbitals interact you get a bonding and an antibonding state. Since each carbon atom contributes one electron you fill the bonding state and leave the anti bonding state empty. This is just your regular covalent bond as in diamond.
By contrast, the remaining $p_z$ orbital interacts with three nearest neighbours, so you'd get (this is all getting a bit arm-waving now!) four bonding and four anti-bonding orbitals. But only four electrons are involved, so only two of the bonding orbitals are filled leaving two empty. When you add in next-nearest neighbours, then next-next-nearest etc the multiple splittings of the bonding orbitals forms an energy band that is only half full. This is your conduction band.
The anti-bonding orbitals will form an empty band some distance above the conduction band. However I believe the bands overlap in graphene so there isn't a band gap.
Needless to say, the electronic structure of graphene is more complicated than the above description suggests, and you shouldn't take the molecular orbital analogy too seriously.