Stacking monolayers of transition metal dichalcogenides into a heterostructure with a finite twist-angle gives rise to artificial moiré superlattices with a tunable periodicity. As a consequence, excitons experience a periodic potential, which can be exploited to tailor optoelectronic properties of these materials. While recent experimental studies have confirmed twist-angle dependent optical spectra, the microscopic origin of moiré exciton resonances has not been fully clarified yet. Here, we combine first principle calculations with the excitonic density matrix formalism to study transitions between different moiré exciton phases and their impact on optical properties of the twisted MoSe2/WSe2 heterostructure. At angles smaller than 2 degrees we find flat, moiré trapped states for inter- and intralayer excitons. This moiré exciton phase drastically changes into completely delocalized states already at 3 degrees. We predict a linear and quadratic twist-angle dependence of excitonic resonances for the moiré-trapped and delocalized exciton phase, respectively. Our work provides microscopic insights opening the possibility to tailor moiré exciton phases in van der Waals superlattices.