Rydberg microwave (MW) sensors are superior to conventional antenna-based techniques because of their wide operating frequency range and outstanding potential sensitivity. Here, we demonstrate a Rydberg microwave receiver with a high sensitivity of $62\,\mathrm{nV} \mathrm{cm}^{-1} \mathrm{Hz}^{-1/2}$ and broad instantaneous bandwidth of up to $10.2\,\mathrm{MHz}$. Such excellent performance was achieved by the amplification of one generated sideband wave induced by the strong coupling field in the six-wave mixing process of the Rydberg superheterodyne receiver, which was well predicted by our theory. Our system, which possesses a uniquely enhanced instantaneous bandwidth and high-sensitivity features that can be improved further, will promote the application of Rydberg microwave electrometry in radar and communication.