Chemical vapor deposition (CVD) has shown great promise for the large-scale production of high-quality graphene films for industrial applications. Atomic-scale theoretical studies can help experiments to deeply understand the graphene growth mechanism, and serve as theoretical guides for further experimental designs. Here, by using density functional theory calculations, ab-initio molecular dynamics simulations, and microkinetic analysis, we systematically investigated the kinetics of hydrogen constrained graphene growth on Cu substrate. The results reveal that the actual hydrogen-rich environment of CVD results in CH as the dominating carbon species and graphene H-terminated edges. CH participated island sp² nucleation avoids chain cyclization process, thereby improving the nucleation and preventing the formation of non-hexameric ring defects. The graphene growth is not simply C-atomic activity, rather, involves three main processes: CH species attachment at the growth edge, leading to a localized sp³ hybridized carbon at the connecting site; excess H transfer from the sp³ carbon to the newly attached CH; and finally dehydrogenation to achieve the sp² reconstruction of the newly grown edge. The threshold reaction barriers for the growth of graphene zigzag (ZZ) and armchair (AC) edges were calculated as 2.46 and 2.16 eV, respectively, thus the AC edge grows faster than the ZZ one. Our theory successfully explained why the circumference of a graphene island grown on Cu substrates is generally dominated by ZZ edges, which is a commonly observed phenomenon in experiments. In addition, the growth rate of graphene on Cu substrates is calculated and matches well with existing experimental observations.