Understanding the microscopic mechanism of chemical vapor deposition (CVD) growth of two-dimensional molybdenum disulfide (2D MoS2) is a fundamental issue towards the function-oriented controlled growth. In this work, we report results on revealing the growth kinetics of 2D MoS2 via capturing the nucleation seed, evolution morphology, edge structure and terminations at the atomic scale during CVD growth using the transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM) studies. The direct growth of few- and mono-layer MoS2 onto graphene based TEM grids allow us to perform the subsequent TEM characterization without any solution-based transfer. Two forms of seeding centers are observed during characterizations: (i) Mo-oxysulfide (MoOxS2-y) nanoparticles either in multi-shelled fullerene-like structures or in compact nanocrystals for the growth of fewer-layer MoS2; (ii) Mo-S atomic clusters in case of monolayer MoS2. In particular, for the monolayer case, at the early stage growth, the morphology appears in irregular polygon shape comprised with two primary edge terminations: S-Mo Klein edge and Mo zigzag edge, approximately in equal numbers, while as the growth proceeds, the morphology further evolves into near-triangle shape in which Mo zigzag edge predominates. Results from density-functional theory calculations are also consistent with the inferred growth kinetics, and thus supportive to the growth mechanism we proposed. In general, the growth mechanisms found here should also be applicable in other 2D materials, such as MoSe2, WS2 and WSe2 etc.