In order to study analytically the nature of the jamming transition in granular material, we have considered a cavity method mean-field theory, in the framework of a statistical mechanics approach, based on Edwards' original idea. For simplicity, we have applied the theory to a lattice model, and a transition with exactly the same nature of the glass transition in mean-field models for usual glass formers is found. The model is also simulated in three dimensions under tap dynamics, and a jamming transition with glassy features is observed. In particular, two-step decays appear in the relaxation functions and dynamic heterogeneities resembling ones usually observed in glassy systems. These results confirm early speculations about the connection between the jamming transition in granular media and the glass transition in usual glass formers, giving moreover a precise interpretation of its nature.