We applied a previously introduced lattice fluid theory for water to study the dissolution of apolar solutes. The treatment is based on a model for the orientation-dependent intermolecular interactions of water. It involves not only the strong directional intermolecular attraction, known as the hydrogen bond, but repulsive interactions between water molecules, which operate at a similar intermolecular separation as hydrogen bonds but at different relative orientations of the molecules, are also important. All the peculiar observations on the dissolution of apolar molecules in water, i.e. temperature dependence of solubility, the Gibbs energy of transfer from apolar medium into water, and its entropic and enthalpic contributions, are reproduced by the theory. Furthermore, the relation between these phenomena and the orientation-dependent intermolecular interactions of water is clarified. Our analysis indicates that the peculiar temperature dependence of the solubility of apolar compounds in water and of the isobaric density of pure water have a common molecular origin, Upon a small expansion of pure water or upon dissolution of small apolar molecules, a subtle enhancement occurs of the type of structuring that is characteristic for water at ambient conditions. This is sufficient to explain the negative hydration entropy of apolar molecules. The negative hydration enthalpy of small apolar molecules is due to a decrease of the number of repulsive non-hydrogen-bonding interactions between neighboring water molecules. Calculations on water adjoining a planar hydrophobic surface indicate that the characteristic features of the dissolution of small apolar solutes do not occur with bulky particles.