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 ∼ 5 × 1039 erg s-1) and radio (L ν ∼ 1029 erg s-1 Hz-1) emission. The radio emission peaked at ∼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 Γβc ≥ 0.55c at ∼100 days. This is faster than the non-relativistic AT 2018cow (Γβc ∼ 0.1c) and closer to ZTF18abvkwla (Γβc ≥ 0.3c at 63 days). The inferred initial kinetic energy of CSS161010 (E k ⪆ 1051 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 (∼ 0.01-0.1 M⊙). 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 ∼150 Mpc in a dwarf galaxy with stellar mass M * ∼ 107 M o˙ and specific star formation rate sSFR ∼ 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.