We present the first angle-resolved photoemission studies of electronic structure in CaC6, a superconducting graphite intercalation compound with T_{c}=11.6 K. We find that, contrary to theoretical models, the electron-phonon coupling on the graphene-derived Fermi sheets with high-frequency graphene-derived phonons is surprisingly strong and anisotropic. The shape of the Fermi surface is found to favor a dynamical intervalley nesting via exchange of high-frequency phonons. Our results suggest that graphene sheets play a crucial role in superconductivity in graphite intercalation compounds.