We present X-ray and radio observations of the Fast Blue Optical Transient CRTS-CSS161010J045834-081803 (CSS161010 hereafter) at t=69-531days. CSS161010 shows luminous X-ray (Lx∼5x1039erg/s) and radio (Lν∼1029erg/s/Hz) emission. The radio emission peaked at ∼100days 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 Γβc≳0.55c at ∼100days. This is faster than the non- relativistic AT 2018cow (Γβc∼0.1c) and closer to ZTF18abvkwla (Γβc≳0.3c at 63days). The inferred initial kinetic energy of CSS161010 (Ek≳1051erg) is comparable to that of long gamma-ray bursts, but the ejecta mass that is coupled to the mildly relativistic outflow is significantly larger (∼0.01-0.1M☉). This is consistent with the lack of observed γ-rays. The luminous X-rays were produced by a different emission component to the synchrotron radio emission. CSS161010 is located at ∼150Mpc in a dwarf galaxy with stellar mass M*∼107M☉ and specific star formation rate sSFR∼0.3/Gyr. 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.