We consider spin injection driven by nonequilibrium chiral phonons from a chiral insulator into an adjacent metal. Phonon-spin conversion arises from the coupling of the electron spin with the microrotation associated with chiral phonons. We derive a microscopic formula for the spin injection rate at a metal-insulator interface. Our results clearly illustrate the microscopic origin of spin current generation by chiral phonons and may lead to a breakthrough in the development of spintronic devices without heavy elements.