Cu substrates are highly preferred for the potential mass production of high-quality graphene, yet many of the important aspects of the atomistic growth mechanisms involved remain to be explored. Using multiscale modeling, we identify C-C dimers as the dominant feeding species in the epitaxial growth of graphene on both Cu(111) and Cu(100) substrates. By contrasting the different activation energies involved in C-C dimer diffusion on terraces and its attachment at graphene island edges, we further reveal why graphene growth is diffusion limited on Cu(111), but attachment limited on Cu(100). We also find even higher potential energy barriers against dimer diffusion along the island edges; consequently, a dendritic-to-compact transition is predicted to take place during graphene enlargement on either substrate, but at different growth temperatures. These findings serve as new insights for better control of epitaxial graphene growth.