The rest mass of a Dirac fermion, such as an electron described by the Dirac equation, is not necessarily zero. In fact, the Dirac equation allows for solutions with both non-zero mass (massive Dirac fermions) and zero mass (massless Dirac fermions). The distinction between massive and massless Dirac fermions is related to the behavior of their energy-momentum dispersion relations.
In the case of graphene, the Dirac cones in the reciprocal space arise due to the unique band structure of the material. Graphene is a two-dimensional honeycomb lattice of carbon atoms, and its electronic structure can be described using a tight-binding model. The tight-binding model leads to a linear dispersion relation near the so-called Dirac points in the reciprocal space, where the conduction and valence bands touch.
The linear dispersion relation near the Dirac points is a consequence of the symmetries of the graphene lattice and the fact that the carbon atoms have a honeycomb arrangement. The low-energy excitations around the Dirac points in graphene are effectively described by massless Dirac fermions. This means that the electrons (or more precisely, the quasiparticles in the material) near the Dirac points behave as if they have zero rest mass, similar to the behavior of massless particles described by the Dirac equation.
However, it's important to note that this behavior is specific to graphene and certain other materials with similar properties. In general, the rest mass of a Dirac fermion is not constrained to be zero. The masslessness of Dirac fermions in graphene is a consequence of the peculiar band structure of the material, rather than a fundamental property of all Dirac fermions.