Graphene, a hexagonal sheet of sp(2)-bonded carbon atoms, has extraordinary properties which hold immense promise for nanoelectronic applications. Unfortunately, the popular preparation methods of micromechanical cleavage and chemical exfoliation of graphite do not easily scale up for application purposes. Epitaxial graphene provides an attractive alternative, though there are many challenges, not least of which is the absence of any understanding of the complex atomistic assembly kinetics of graphene layers. Here, we present a simple rate theory of epitaxial graphene growth on close-packed metal surfaces. On the basis of recent low-energy electron-diffraction microscopy experiments, our theory supposes that graphene islands grow predominantly by the attachment of five-atom clusters. With optimized kinetic parameters, our theory produces a quantitative account of the measured time-dependent carbon adatom density. The temperature dependence of this density at the onset of nucleation leads us to predict that the smallest stable precursor to graphene growth is an immobile island composed of six five-atom clusters. This conclusion is supported by a recent study based on temperature-programmed growth of epitaxial graphene, which provides direct evidence of nanoclusters whose coarsening leads to the formation of graphene layers. Our findings should motivate additional high-resolution imaging experiments and more detailed simulations which will yield important input to developing strategies for the large-scale production of epitaxial graphene.