Recent experiments have shown that electric double-layer capacitors utilizing electrodes consisting of graphene edge plane exhibit higher capacitance than graphene basal plane. However, theoretical understanding of this capacitance enhancement is still limited. Here we applied a self-consistent joint density functional theory calculation on the electrode/electrolyte interface and found that the capacitance of graphene edge plane depends on the edge type: zigzag edge has higher capacitance than armchair edge due to the difference in their electronic structures. We further examined the quantum, dielectric, and electric double-layer (EDL) contributions to the total capacitance of the edge-plane electrodes. Classical molecular dynamics simulation found that the edge planes have higher EDL capacitance than the basal plane due to better adsorption of counter-ions and higher solvent accessible surface area. Our work therefore has elucidated the capacitive energy storage in graphene edge planes that take into account both the electrode's electronic structure and the EDL structure.