Many-body interactions between dynamical agents have caught particular attention in recent works that found wide applications in physics, neuroscience, and sociology. In this paper we investigate such higher order (nonadditive) interactions on collective dynamics in a system of globally coupled heterogeneous phase oscillators. We show that the three-body interactions encoded microscopically in nonlinear couplings give rise to added dynamic phenomena occurring beyond the pairwise interactions. The system in general displays an abrupt desynchronization transition characterized by irreversible explosive synchronization via an infinite hysteresis loop. More importantly, we give a mathematical argument that such an abrupt dynamic pattern is a universally expected effect. Furthermore, the origin of this abrupt transition is uncovered by performing a rigorous stability analysis of the equilibrium states, as well as by providing a detailed description of the spectrum structure of linearization around the steady states. Our work reveals a self-organized phenomenon that is responsible for the rapid switching to synchronization in diverse complex systems exhibiting critical transitions with nonpairwise interactions.