Single-photon sources are an integral part of various quantum technologies, and solid-state quantum emitters at room temperature appear to be a promising implementation. We couple the fluorescence of individual silicon-vacancy centers in nanodiamonds to a tunable optical microcavity to demonstrate a single-photon source with high efficiency, increased emission rate, and improved spectral purity compared to the intrinsic emitter properties. We use a fiber-based microcavity with a mode volume as small as and a quality factor of and observe an effective Purcell factor of up to 9.2. Furthermore, we study modifications of the internal rate dynamics and propose a rate model that closely agrees with the measurements. We observe lifetime changes of up to 31%, limited by the finite quantum efficiency of the emitters studied here. With improved materials, our achieved parameters predict single-photon rates beyond 1 GHz.