Chromium triiodide is an intrinsically magnetic van der Waals material down to the single-layer limit. Here, we provide a first-principles description of finite-temperature magnetic and spectral properties of monolayer (ML) CrI3 based on fully charge self-consistent density functional theory (DFT) combined with dynamical mean-field theory, revealing a formation of local moments on Cr from strong local Coulomb interactions. We show that the presence of local dynamical correlations leads to a modification of the electronic structure of ferromagnetically ordered CrI3. In contrast to conventional DFT+U calculations, we find that the top of the valence band in ML CrI3 demonstrates essentially different orbital character for minority and majority spin states, which is closer to the standard DFT results. This leads to a strong spin polarization of the optical conductivity upon hole doping, which could be verified experimentally.