The finding that electronic conductance across ultrathin protein films between metallic electrodes remains nearly constant from room temperature to just a few degrees Kelvin has posed a challenge. We show that a model based on a generalized Landauer formula explains the nearly constant conductance and predicts an Arrhenius-like dependence for low temperatures. A critical aspect of the model is that the relevant activation energy for conductance is either the difference between the HOMO and HOMO-1 or the LUMO+1 and LUMO energies instead of the HOMO-LUMO gap of the proteins. Analysis of experimental data confirms the Arrhenius-like law and allows us to extract the activation energies. We then calculate the energy differences with advanced DFT methods for proteins used in the experiments. Our main result is that the experimental and theoretical activation energies for these three different proteins and three differently prepared solid-state junctions match nearly perfectly, implying the mechanism's validity.