Wafer-scale single-crystal graphene film directly grown on insulating substrates via the chemical vapor deposition (CVD) method is desired for building high-performance graphene-based devices. In comparison with the well-studied mechanism of graphene growth on transition metal substrates, the lack of understanding on the mechanism of graphene growth on insulating surfaces greatly hinders the progress. Here, by using first-principles calculation, we systematically explored the absorption of various carbon species CHx (x = 0, 1, 2, 3, 4) on three typical insulating substrates [h-BN, sapphire, and quartz] and reveal that graphene growth on an insulating surface is dominated by the reaction of active carbon species with the hydrogen-passivated graphene edges and thus is less sensitive to the type of the substrate. The dominating gas phase precursor, CH3, plays two key roles in graphene CVD growth on an insulating substrate: (i) to feed the graphene growth and (ii) to remove excessive hydrogen atoms from the edge of graphene. The threshold reaction barriers for the growth of graphene armchair (AC) and zigzag (ZZ) edges were calculated as 3.00 and 1.94 eV, respectively; thus the ZZ edge grows faster than the AC one. Our theory successfully explained why the circumference of a graphene island grown on insulating substrates is generally dominated by AC edges, which is a long-standing puzzle of graphene growth. In addition, the very slow graphene growth rate on an insulating substrate is calculated and agrees well with existing experimental observations. The comprehensive insights on the graphene growth on insulating surfaces at the atomic scale provide guidance on the experimental design for high-quality graphene growth on insulating substrates.