The observed distribution of neutron star (NS) masses reflects the physics of core-collapse supernova explosions and the structure of the massive stars that produce them at the end of their evolution. We present a Bayesian analysis that directly compares the NS mass distribution observed in double NS systems to theoretical models of NS formation. First, we find that models with standard binary mass ratio distributions are strongly preferred over independently picking the masses from the initial mass function, although the strength of the inference depends on whether current assumptions for identifying the remnants of the primary and secondary stars are correct. Secondly, NS formation models with no mass fallback are favoured because they reduce the dispersion in NS masses. The double NS system masses thus directly point to the mass coordinate where the supernova explosion was initiated, making them an excellent probe of the supernova explosion mechanism. If we assume no fallback and simply vary the mass coordinate separating the remnant and the supernova ejecta, we find that for solar metallicity stars the explosion most likely develops at the edge of the iron core at a specific entropy of S/NA≈ 2.8 kB. The primary limitations of our study are the poor knowledge of the supernova explosion mechanism and the lack of broad range of supernova model explosions of Large Magellanic Cloud to solar metallicity.