Monte Carlo simulations are used to investigate the mechanism of the disorder-to-order phase transition for a bulk system of colloidal hard cubes. It is observed that the structure of the ordered state is foreshadowed in the disordered state through multiple spontaneously occurring ordered domains. Such domains arise due to the entropic preference for local facet alignment between particles and occur transiently and sparsely throughout the system even in the stable isotropic phase. At pressures (and degrees of supersaturation) where the isotropic phase becomes marginally metastable, a classical nucleation process is never observed; instead, the ordered domains increase in number and size, eventually reaching a critical point where they percolate the entire system and spontaneously consolidate to form the ordered phase. The critical number of particles and the per particle free-energy barrier both decrease with pressure. Using the total number of locally ordered particles as a global order parameter, it is predicted that for large systems the ordering transition would only be spontaneous above a critical pressure. Finally, a test designed to probe the ability of the system to favor a single monodomain solid from initially misaligned-ordered domains, reveals that an active interdomain zone mediates the concerted reorientation of particles.