The recent observation of room temperature spin-dependent photoluminescence (PL) emission from hexagonal boron nitride's (h-BN's) defect centers motivates for performing a complementary low-temperature photophysical study of quantum emitters under relatively high magnetic fields. Here, we investigate the PL emission dynamics of h-BN's visible single-photon emitters under an applied out-of-plane magnetic field at cryogenic temperatures. The PL intensity of the emitters in our work strikingly exhibits strong magnetic field dependence and decreases with the increased magnetic field. A substantial decrease in the integrated PL intensity of the emitters by up to one order of magnitude was observed when the applied field is increased from 0 T to 7 T. The observed reversible photodarkening of PL emission due to the applied magnetic field is in very well agreement with the predictions of a recent joint experimental and theoretical study and can happen only if the spin-selective, non-radiative, and asymmetric intersystem crossing transitions proceed from the triplet excited state to the lowest-lying spin-singlet metastable state and from the metastable state to the triplet ground state. Our results not only shed more light on the light emission paths of defect centers in h-BN but also show the use of the magnetic field as an efficient control knob in the development of magneto-optical devices.