Using a combination of spectroscopic ellipsometry and DC transport measurements, we determine the temperature dependence of the optical conductivity of NdNiO3 and SmNiO3 films. The optical spectra show the appearance of a characteristic two-peak structure in the near-infrared when the material passes from the metal to the insulator phase. Dynamical mean-field theory calculations confirm this two-peak structure and allow us to identify these spectral changes and the associated changes in the electronic structure. We demonstrate that the insulating phase in these compounds and the associated characteristic two-peak structure are due to the combined effect of bond disproportionation and Mott physics associated with half of the disproportionated sites. We also provide insights into the structure of excited states above the gap.