The stress response of permanently crosslinked gelatin gels was recently observed to display glass-like features, namely, a stretched-exponential behavior terminated by an exponential decay, the characteristic time scales of which increase dramatically with decreasing temperature. This phenomenon is studied here using a model of flexible polymer gel network where relaxation proceeds via elementary monomer exchanges between helix and coil segments. The relaxation dynamics of a full network simulation is found to be nearly identical to that of a model of independent strands, which shows that for flexible polymer gels in the range of elastic moduli of interest, both strand contour length disorder and elastic couplings are irrelevant. We thus focus on the independent strand model and find it not only to explain the observed functional form of the stress relaxation curves but also to yield predictions that match very satisfactorily the experimental measurements of final relaxation time and total stress drop. The system under study thus constitutes a rare case where the origin of glass-like behavior can be unambiguously identified, namely, as the signature of the enhancement of helix content fluctuations when approaching from above the mean-field helix-coil transition of strands.