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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) CrI$_3$ 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 CrI$_3$. In contrast to conventional DFT+$U$ calculations, we find that the top of the valence band in ML CrI$_3$ 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.