Emergence of new protein structures has proved difficult to trace in nature and engineer in the laboratory. However, one aspect of structure evolution has proved immensely helpful for determining the three-dimensional structure of proteins from their sequences: in the vast majority of cases, proteins that share more than 30% sequence identity have similar structures. Below this mark is the "twilight zone" where proteins may have identical or very different structures. These observations form the foundational intuition behind structure homology modeling. Despite their importance, however, they have never received a comprehensive biophysical justification. Here we show that the onset of the twilight zone is more gradual for proteins with low contact density, a proxy for low thermodynamic stability, than proteins with high contact density. Then we present an analytical model that treats divergent fold evolution as an activated process, in analogy to chemical kinetics, where sequence evolution must overcome thermodynamically unstable evolutionary intermediates to discover new folds. This model explains the existence of a twilight zone and explains why its onset is more abrupt for some classes of proteins than for others. We test the assumptions of the model and characterize the dynamics of fold evolution using evolutionary simulations of model proteins and cell populations. Overall these results show how fundamental biophysical constraints directed evolutionary dynamics leading to the Universe of modern protein structures and sequences.