The freshwater polyp Hydra has considerable regeneration capabilities. A small fragment of tissue excised from an adult animal is sufficient to regenerate an entire Hydra in the course of a few days. During the initial stages of the regeneration process, the tissue forms a hollow sphere. Then the sphere exhibits shape oscillations in the form of repeated cycles of swelling and collapse. We propose a biophysical model for the swelling mechanism. Our model takes the osmotic pressure difference between Hydra's inner and outer media and the elastic forces of the Hydra shell into account. We validate the model by a comprehensive experimental study including variations in initial medium concentrations, Hydra sphere sizes and temperatures. Numerical simulations of the model provide values for the swelling rates that are in agreement with the ones measured experimentally. Based on our results we argue that the shape oscillations are a consequence of Hydra's osmoregulation.