Biomembranes are key objects of numerous studies in biology and biophysics of great importance to medicine. A few nanometers thin quasi two-dimensional liquid crystalline membranes with bending rigidity of a few kT exhibit unusual properties and they are the focus of theoretical and experimental physics. The first order chain-melting phase transition of lipid membranes is observed to be accompanied by a pseudocritical behavior of membrane physical-chemical properties. However, the investigation of the nature of the anomalous swelling of a stack of lipid membranes in the vicinity of the transition by different groups led to conflicting conclusions about the level of critical density fluctuations and their impact on the membrane softening. Correspondingly, conclusions about the contribution of Helfrich's undulations to the effect of swelling were different. In our work we present a comprehensive complementary neutron small-angle and spin-echo study directly showing the presence of significant critical fluctuations in the vicinity of the transition which induce membrane softening. However, contrary to the existing paradigm, we demonstrate that the increased undulation forces cannot explain the anomalous swelling. We suggest that the observed effect is instead determined by the dominating increase of short-range entropic repulsion.