The Large Binocular Telescope Interferometer (LBTI) is a versatile instrument designed for highangular resolution and high-contrast infrared imaging (1.5-13 µm). In this paper, we focus on the mid-infrared (8-13 µm) nulling mode and present its theory of operation, data reduction, and onsky performance as of the end of the commissioning phase in March 2015. With an interferometric baseline of 14.4 meters, the LBTI nuller is specifically tuned to resolve the habitable zone of nearby main-sequence stars, where warm exozodiacal dust emission peaks. Measuring the exozodi luminosity function of nearby main-sequence stars is a key milestone to prepare for future exoEarth direct imaging instruments. Thanks to recent progress in wavefront control and phase stabilization, as well as in data reduction techniques, the LBTI demonstrated in February 2015 a calibrated null accuracy of 0.05% over a three-hour long observing sequence on the bright nearby A3V star β Leo. This is equivalent to an exozodiacal disk density of 15 to 30 zodi for a Sun-like star located at 10 pc, depending on the adopted disk model. This result sets a new record for high-contrast mid-infrared interferometric imaging and opens a new window on the study of planetary systems.