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Lipid vesicles have received considerable interest because of their applications to in vitro reductionist cell membrane models as well as therapeutic delivery vehicles. In these contexts, the mechanical response of vesicles in nonequilibrium environments plays a key role in determining the corresponding dynamics. A common understanding of the response of lipid vesicles upon exposure to a hypotonic solution is a characteristic pulsatile behavior. Recent experiments, however, have shown vesicles exploding under an osmotic shock generated by photo-reactions, yet the explanatory mechanism is unknown. Here we present a generalized biophysical model incorporating a stochastic account of membrane rupture to describe both swell-burst-reseal cycling and exploding dynamics. This model agrees well with experimental observations, and it unravels that the sudden osmotic shock strains the vesicle at an extreme rate, driving the vesicle into buckling instabilities responsible for membrane fragmentation, i.e. explosion. Our work not only advances the fundamental framework for non-equilibrium vesicle dynamics under osmotic stress, but also offers design guidelines for programmable vesicle-encapsulated substance release in therapeutic carriers.