Disordered noble-metal nanoparticle films exhibit highly localized and stable non-linear light emission from sub-diffraction regions upon illumination by near-infrared femtosecond pulses. Such hot spot emission spans a continuum in the visible and near-infrared spectral range. Strong plasmonic enhancement of light-matter interaction and the resulting complexity of experimental observations have prevented the development of a universal understanding of the origin of light emission. Here, we study the dependence of emission spectra on excitation irradiance and provide the most direct evidence yet that the continuum emission observed from both silver and gold nanoparticle aggregate surfaces is caused by recombination of hot electrons within the conduction band. The electron gas in the emiting particles, which is effectively decoupled from the lattice temperature for the duration of emission, reaches effective temperatures of several thousand Kelvin and acts as a sub-diffraction incandescent light source on sub-picosecond time scales.