Don't think about annihilation as something exceptional. Annihilation is just a type of interaction and there are many other possible interactions. Now I don't use the word "interaction" in the sense of "4 fundamental interactions", but in the sense of possible process in the quantum world where particles are destroyed and created. So the question should rather be - why there are interactions?
Annihilation is simply one of the processes allowed by the interaction term in the Lagrangian. See for example the interaction part of the QED Lagrangian:
Notice that the term couples the Dirac (for instance electron) field with the electromagnetic field. All possible processes follow from that: you can have electron emitting a photon, photon absorbed by electron, collison of 2 electrons creating a photon, collison of an electron with a positron creating 2 photons (this is what we call annihilation), but such collision can also create heavier particles (like muon and antimuon).
Annihilation is just the case where the quantum numbers cancel. But there is nothing really special about it. We should expect that there will be certain pairs of particles which have cancelling quantum numbers and other pairs that don't.