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 behaviour is described by the general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behaviour that defies this description and is not well understood. A paradigmatic example is the many-body localization (MBL) transition, which marks the breakdown of thermalization in an isolated quantum many-body system as its disorder increases beyond a critical value. Characterizing quantum critical behaviour in an MBL system requires probing its entanglement over space and time, which has proved experimentally challenging owing to stringent requirements on quantum state preparation and system isolation. We observed quantum critical behaviour at the MBL transition in a disordered Bose–Hubbard system and characterized its entanglement via its multi-point quantum correlations. We observed the emergence of strong correlations, accompanied by the onset of anomalous diffusive transport throughout the system, and verified their critical nature by measuring their dependence on the system size. 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 connect the macroscopic phenomenology of the transition to the system’s microscopic structure of quantum correlations, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems.