Phase transitions are driven by collective fluctuations of a system’s constituents that emerge at a critical point. This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behavior is described by a general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behavior that defies this description and is not well understood. A paradigmatic example is the many-body-localization (MBL) transition, which marks the breakdown of quantum thermalization. Characterizing quantum critical behavior in an MBL system requires the measurement of its entanglement properties over space and time, which has proven experimentally challenging due to stringent requirements on quantum state preparation and system isolation. Here, we observe quantum critical behavior at the MBL transition in a disordered Bose-Hubbard system and characterize its entanglement properties via its quantum correlations. We observe strong correlations, whose emergence is accompanied by the onset of anomalous diffusive transport throughout the system, and verify their critical nature by measuring their system-size dependence. The correlations extend to high orders in the quantum critical regime and appear to form via a sparse network of many-body resonances that spans the entire system. Our results unify the system’s microscopic structure with its macroscopic quantum critical behavior, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems.