Dense particulate suspensions exhibit a dramatic increase in average viscosity above a critical, material-dependent shear stress. This thickening changes from continuous to discontinuous as the concentration is increased. Using direct measurements of spatially resolved surface stresses in the continuous thickening regime, we report the existence of clearly defined dynamic localized regions of substantially increased stress that appear intermittently at stresses above the critical stress. With increasing applied stress, these regions occupy an increasing fraction of the system, and the increase accounts quantitatively for the observed shear thickening. The regions represent high-viscosity fluid phases, with a size determined by the distance between the shearing surfaces and a viscosity that is nearly independent of shear rate but that increases rapidly with concentration. Thus, we find that continuous shear thickening arises from increasingly frequent localized discontinuous transitions between distinct fluid phases with widely differing viscosities.