We investigate the molecular mechanism of pressure denaturation of proteins using the angle-dependent integral equation theory combined with the multipole water model and the morphometric approach. We argue that the hydration entropy of a protein is the key quantity. It is verified that at an elevated pressure, a swelling structure--which has only moderately less compact than the native structure but has a much larger water-accessible surface area--turns more stable than the native structure in terms of the water entropy. The swelling structure is characterized by the penetration of water into the interior. The hydration entropy is decomposed into contributions from the translational and rotational restrictions for the molecular motions of water. Each contribution is further decomposed into the water-protein pair correlation component and the water-water-protein triplet and higher-order correlation components. The pair correlation component in the translational contribution is divided into two terms arising from the excluded volume and the water structure near the protein, respectively. It is found that pressure denaturation accompanies a loss of the translational and rotational entropies at the pair correlation level but a much larger gain of the translational entropy at the triplet and higher-order correlation levels. Although the translational and rotational motions of water molecules penetrating the protein interior and contacting the protein surface are constrained, the translational restriction for the water molecules well outside the protein is greatly reduced. The latter entropic gain dominates, leading to the denaturation.