We present X-ray and radio observations of the Fast Blue Optical Transient CRTS-CSS161010 J045834-081803 (CSS161010 hereafter) at t = 69-531 days. CSS161010 shows luminous X-ray (L-x similar to 5 x 10(39) erg s(-1)) and radio (L-nu similar to 10(29) erg s(-1) Hz(-1)) emission. The radio emission peaked at similar to 100 days post-transient explosion and rapidly decayed. We interpret these observations in the context of synchrotron emission from an expanding blast wave. CSS161010 launched a mildly relativistic outflow with velocity Gamma beta c >= 0.55c at similar to 100 days. This is faster than the non-relativistic AT 2018cow (Gamma beta c similar to 0.1c) and closer to ZTF18abvkwla (Gamma beta c >= 0.3c at 63 days). The inferred initial kinetic energy of CSS161010 (E-k greater than or similar to 10(51) erg) is comparable to that of long gamma-ray bursts, but the ejecta mass that is coupled to the mildly relativistic outflow is significantly larger (similar to 0.01-.1 M-circle dot). This is consistent with the lack of observed gamma-rays. The luminous X-rays were produced by a different emission component to the synchrotron radio emission. CSS161010 is located at similar to 150 Mpc in a dwarf galaxy with stellar mass M-* similar to 10(7) M-circle dot and specific star formation rate sSFR similar to 0.3 Gyr(-1). This mass is among the lowest inferred for host galaxies of explosive transients from massive stars. Our observations of CSS161010 are consistent with an engine-driven aspherical explosion from a rare evolutionary path of a H-rich stellar progenitor, but we cannot rule out a stellar tidal disruption event on a centrally located intermediate-mass black hole. Regardless of the physical mechanism, CSS161010 establishes the existence of a new class of rare (rate < 0.4% of the local core-collapse supernova rate) H-rich transients that can launch mildly relativistic outflows.