Ok, it is not really the material that is exciton-based, but rather applications.
Excitons are mainly important for the optical properties of a material. Usually optical recombination causing the emission of photons from a semiconductor occurs between free electrons and holes. For both there is a continuum of possible energies and therefore you will also get a rather broad range of photon emission energies. Now if excitons are formed, this means that there is a bound electron-hole state at an energy slightly lower than the band gap energy - it is in fact reduced by the amount of the exciton binding energy. Although also excitons can in principle have kinetic energy and therefore a broader spectrum, it is not as broad as the free-carrier spectrum. Also you have the possibility to trap excitons and reduce their motion by using low-dimensional nanostructures like quantum wells or quantum dots which further narrows the energy spread.
Such a narrow energy spread is highly attractive for optical devices. It is much easier to couple to or build lasers on a material which has a rather discrete density of states than on one which shows a broad continuum.