Low-temperature photoluminescence (PL) of hBN-encapsulated monolayer tungsten diselenide (WSe2) shows a multitude of sharp emission peaks below the bright exciton. Some of them have been recently identified as phonon sidebands of momentum-dark states. However, the exciton dynamics behind the emergence of these sidebands has not been revealed yet. In this joint theory-experiment study, we theoretically predict and experimentally observe time-resolved PL, providing microscopic insights into the thermalization of hot excitons formed after optical excitation. In very good agreement between theory and experiment, we demonstrate a spectral red-shift of phonon sidebands on a time scale of tens of picoseconds, reflecting the phonon-driven thermalization of hot excitons in momentum-dark states. Furthermore, we predict the emergence of a transient phonon sideband that vanishes in the stationary PL. The obtained microscopic insights are applicable to a broad class of 2D materials with multiple exciton valleys.