The 0th Landau level also has a degeneracy of four as do other LLs. The significance of 0th LL is that it is shared by conduction and valence band with equal weight. That is to say: two of 0th LLs are electron like and two of them are hole-like (because the degeneracy is four, I imagine there are four "seperated" non-degenerate LLs at the same energy for pedagogical simplicity).
When an electron-like Landau-level is fully occupied by electrons, it contributes a unit conductance; when a hole-like Landau-level is fully occupied by holes, it contributes a negative unit conductance. (Assuming that the landau level has no degeneracy)
When an electron-like Landau-level is empty, it contributes no conductance (of course!); when a hole-like Landau-level is filled with electrons, it contributes no conductance.
Now let's see how the anomalous quantum hall effect arises:
When the 0th LL is fully occupied, the two electron-like LLs are filled with electron and two hole-like LLs are also filled with electron. Thus the conductance of the system is 2 unit conductance.
When the 0th LL is empty, the two electron-like LLs are empty and two hole-like LLs are filled with holes. Thus contribute to the conductance of -2 unit conductance.
Similarly, when both 0th and 1th LLs are also occupied with electrons, the total conductance is 4+2.
The origin of the anomalous Hall conductance
The anomalous Hall conductance arises because of the particle hole symmetry and the linear dispersion of graphene. For the 0th LL, because its energy is just 0, so it must be shared by the valence and conduction band.
However, for the parabolic dispersion, even if the original band have particle-hole symmetry. When forming LLs, the energy of lowest LLs formed by conduction band is 1/2, While the energy of lowest LL formed by valence band is -1/2. They do not mix.