We present a comprehensive theoretical study of conventional superconductivity in cubic antiperovskites materials with composition XYZ₃ where X and Z are metals and Y is H, B, C, N, O, and P. Our starting point are electron-phonon calculations for 384 materials performed with density-functional perturbation theory. While 40% of the materials were dynamically unstable as they exhibited imaginary frequencies, we discovered 16 compounds with Tc higher than 5 K including antiperovskites with Y=H, N, C and O. We used these results to train interpretable machine learning models to understand and further explore this family of compounds. This lead us to predict a further 44 materials with superconducting transition temperatures above 5 K, reaching a maximum of 17.8 K for PtHBe₃. Furthermore, the models give us an understanding of the mechanism of superconductivity in anti-perovskites and highlight the importance of the density of states at the Fermi level and of the mass of the Y-atom for the strength of the phonon coupling. Finally, we study in detail a few systems, uncovering some issues with previously published theoretical data. The combination of traditional approaches with interpretable machine learning turns out to be a very efficient methodology to study and systematize whole classes of materials, and is easily extendable to other families of compounds or physical properties.