If we have a bunch of electrons in the conduction band, there's a probability that some will fall into valence band. When that happens, it's a recombination event - i.e. the a hole is filled and disappears. The more electrons you have, the more falling electrons you have. So if the number of electrons goes up, the number of holes must go down (they keep getting killed off by recombination).
The same argument applies to having a lot of holes. With so many empty states (holes) available, electrons find it easy to find a hole to fall into, so they get killed off too. Alternatively, you can think of holes "falling up" because they're positively charged and band diagrams are in negative energy (i.e. electron energy). When they fall up, they meet an electron and both go away.
So whenever one goes up, the other must go down because the chances of holes and electrons "meeting" each other increases. In intrinsic semiconductors this means their numbers are equal: n = p = ni. Donors or acceptors imbalance this, but np = ni^2 still holds until degeneracy happens. To see this relation, write n, p, and ni in terms Nc and Nv.
