The unusually short durations, high luminosities, and high photon energies of the Crab Nebula gamma-ray flares require relativistic bulk motion of the emitting plasma. We explain the Crab flares as the result of randomly oriented relativistic "minijets" originating from reconnection events in a magnetically dominated plasma. We develop a statistical model of the emission from Doppler boosted reconnection minijets and find analytical expressions for the moments of the resulting nebula light curve (e.g. time average, variance, skewness). The light curve has a flat power spectrum that transitions at short timescales to a decreasing power-law of index 2. The flux distribution from minijets follows a decreasing power-law of index ~ 1, implying the average flux from flares is dominated by bright rare events. The predictions for the flares' statistics can be tested against forthcoming observations. We find the observed flare spectral energy distributions (SEDs) have several notable features: A hard power-law index of p ~< 1 for accelerated particles that is expected in various reconnection models, including some evidence of a pile-up near the radiation reaction limit. Also, the photon energy at which the SED peaks is higher than that implied by the synchrotron radiation reaction limit, indicating the flare emission regions' Doppler factors are ~> few. We conclude that magnetic reconnection can be an important, if not dominant, mechanism of particle acceleration within the nebula.