29 pages, 7 figs, ; We investigate the dynamics of kinetically constrained models of glassformers by analysing the statistics of trajectories of the dynamics, orhistories, using large deviation function methods. We show that, in general,these models exhibit a first-order dynamical transition between active andinactive dynamical phases. We argue that the dynamical heterogeneitiesdisplayed by these systems are a manifestation of dynamical first-order phasecoexistence. In particular, we calculate dynamical large deviation functions,both analytically and numerically, for the Fredrickson-Andersen model, the Eastmodel, and constrained lattice gas models. We also show how large deviationfunctions can be obtained from a Landau-like theory for dynamical fluctuations.We discuss possibilities for similar dynamical phase-coexistence behaviour inother systems with heterogeneous dynamics.