The answer lies in an understanding of what charge really is.
i.e. a fundamental force carried by some sub-atomic particles, the most familiar of these are the electron and proton.
Unlike charges attract and like charge repel. The charge force is measured to a high degree of accuracy as is the mass of the particles that carry it. So far so good.
All materials acquire their physical properties from the interaction of these charge carrying particles.
In a rigid material, protons (+ve charge) form part of an atoms nucleus, are fixed in relation to other nuclei and not free to move.
Electrons orbit the nucleus in specific shells. However, some atoms have loosely bound electrons in their outer shells which are free to be dislodged and can move through the bulk material accordingly. Most notably, these are conductors and semi-conductors and the plates of a capacitor are constructed from conductors.
When a build up of electrons occurs on one plate of a capacitor, the electrons on the other plate will exerience a replusive force (like charges repel) and will try and migrate as far as possible from the source of the repulsive force. As they do so, the fixed position atoms they leave behind will have a net +ve charge because the -ve charge electrons have migrated away leaving an excess of +ve charge protons.
Hence, the plate with the excess electrons will be net -ve charge and the plate with the excess protons will be net +ve charge.
The balance comes from the fact that the charge force carried by both protons and electrons are identical in magnitude but opposite in polarity (force vector) and therefore reach a state of equilibrium.
If you can say how many atoms exactly there are in each plate, moreover, that BOTH plates have the exact same number of atoms, you have your answer.