We present a theory of current conduction through buckyball (C(60)) molecules on silicon by coupling a density functional treatment of the molecular levels embedded in a semiempirical treatment of the silicon surface with a nonequilibrium Green's function treatment of quantum transport. Several experimental variations in conductance-voltage characteristics are quantitatively accounted for by varying the detailed molecule-silicon bonding geometries. We identify how variations in contact surface microstructure influence the number, positions, and shapes of the conductance peaks, while varying separations of the scanning probe from the molecules influence their peak amplitudes.