The recent transient event Swift J1644+57 has been interpreted as resulting from a relativistic outflow, powered by the accretion of a tidally disrupted star onto a supermassive black hole. This discovery of a new class of relativistic transients opens new windows into the study of tidal disruption events (TDEs) and offers a unique probe of the physics of relativistic jet formation and the conditions in the centers of distant quiescent galaxies. Unlike the rapidly-varying γ/X-ray emission from Swift J1644+57, the radio emission varies more slowly and is well modeled as synchrotron radiation from the shock interaction between the jet and the gaseous circumnuclear medium (CNM). Early after the onset of the jet, a reverse shock propagates through and decelerates the ejecta released during the first few days of activity, while at much later times the outflow approaches the self-similar evolution of Blandford and McKee. The point at which the reverse shock entirely crosses the earliest ejecta is clearly observed as an achromatic break in the radio light curve at t ≈ 10 days. I discuss the implications of Swift J1644+57 for the fraction of TDEs accompanied by relativistic jets; the physics of jet formation more broadly; and the prospects for detecting off-axis TDE radio emission, either via follow-up observations of TDE candidates discovered at other wavelengths or blindly with upcoming wide-field radio surveys.