Luminous blue variables are massive, evolved stars that exhibit large variations in luminosity and size on timescales from months to years, with high associated rates of mass loss1-5. In addition to this on-going variability, these stars exhibit outburst phases, during which their size increases and as a result their effective temperature decreases, typically to about 9,000 kelvin3,6. Outbursts are believed to be caused by the radiation force on the cooler, more opaque, outer layers of the star balancing or even exceeding the force of gravity, although the exact mechanisms are unknown and cannot be determined using one-dimensional, spherically symmetric models of stars because such models cannot determine the physical processes that occur in this regime7. Here we report three-dimensional simulations of massive, radiation-dominated stars, which show that helium opacity has an important role in triggering outbursts and setting the observed effective temperature during outbursts of about 9,000 kelvin. It probably also triggers the episodic mass loss at rates of 10-7 to 10-5 solar masses per year. The peak in helium opacity is evident in our three-dimensional simulations only because the density and temperature of the stellar envelope (the outer part of the star near the photosphere) need to be determined self-consistently with convection, which cannot be done in one-dimensional models that assume spherical symmetry. The simulations reproduce observations of long-timescale variability, and predict that convection causes irregular oscillations in the radii of the stars and variations in brightness of 10-30 per cent on a typical timescale of a few days. The amplitudes of these short-timescale variations are predicted to be even larger for cooler stars (in the outburst phase). This short-timescale variability should be observable with high-cadence observations.