Color centers in diamond provide a promising platform for quantum optics in the solid state, with coherent optical transitions and long-lived electron and nuclear spins. Building upon recent demonstrations of nanophotonic waveguides and optical cavities in single-crystal diamond, we now demonstrate on-chip diamond nanophotonics with a high efficiency fiber-optical interface, achieving > 90% power coupling at visible wavelengths. We use this approach to demonstrate a bright source of narrowband single photons, based on a silicon-vacancy color center embedded within a waveguide-coupled diamond photonic crystal cavity. Our fiber-coupled diamond quantum nanophotonic interface results in a high flux of coherent single photons into a single mode fiber, enabling new possibilities for realizing quantum networks that interface multiple emitters, both on-chip and separated by long distances.