Using both coarse-grained (CG) and fine-grained (FG) simulations we show how strain hardening in polymeric glasses under uniaxial extension arises from highly stretched strands that form as the polymer chains deform subaffinely on increasing length scales as strain increases. The coarse-grained simulations are performed using the hybrid Brownian dynamics method (HBD) [Zou, W.; Larson, R G. Soft Matter 2016, 3, 3853-3865] with 10-30 coarse-grained springs per polymer chain, while the fine-grained simulations employ the Kremer-Grest bead-spring model with 600 beads per chain. We find that the HBD model accurately predicts how the MD chain configurations evolve during deformation despite being a single-chain-in-mean-field model that does not account for entanglements or monomer-level structure. We show using both models that the glassy strain hardening modulus G(R) is much larger than the melt plateau modulus G(N) because chain segments become highly stretched at modest Hencky strain (epsilon < similar to 1) owing to the high interchain friction in the glass. HBD model predictions of strain hardening match those of the MD simulations in shape and magnitude, relative to the flow stress, which is the stress just beyond the yield point, for several deformation protocols, and also capture the increase in strain hardening with increasing chain length that saturates in the long chain limit. As deformation proceeds, chains begin to form kinks or folds (starting at a Hencky strain epsilon approximate to 1.0) analogous to those produced in extensional flows of dilute and entangled polymer solutions. We identify "entangled kinks" in the MD simulations; these do not appear to strongly influence strain hardening but may be important in delaying fracture. Motivated by these results, we improve upon HBD's ability to accurately capture stress-strain curves at small strains through yielding and strain softening by extending the theory to multiple segmental relaxation modes, whose strain-dependent relaxation times are obtained from small-molecule probe relaxation experiments by Ediger and co-workers [Bending, B.; Ediger, M. D. J. Polym. Sci. B 2016, 54, 1957-1967]. This produces excellent agreement between the HBD model and experimental stress-strain curves through the yield point but requires segmental relaxation data for each experiment. Future work should aim at developing a constitutive equation for the segmental relaxation.