The origin of compression-induced failure in brittle solids has been a subject of debate. Using in situ, high-speed, strain field mapping of a representative material, polymethylmethacrylate, we reveal that shock loading leads to heterogeneity in compressive strain field, which in turn gives rise to localized lateral tension and shear through Poisson's effects, and subsequently, localized microdamage. A failure wave nucleates from the impact surface and its propagation into the microdamage zone is self-sustained, and triggers interior failure. Its velocity increases with increasing shock strength and eventually approaches the shock velocity. The seemingly puzzling phenomena observed in previous experiments, including incubation time, failure wave velocity variations, and surface roughness effects, can all be explained consistently with nucleation and growth of microdamage, and the effects of loading strength and preexisitng defects.