The hydrophobic effect (HE) is commonly associated with the demixing of oil and water at ambient conditions and plays the leading role in determining the structure and stability of biomolecular assembly in aqueous solutions. On the molecular scale HE has an entropic origin. It is believed that hydrophobic particles induce order in the surrounding water by reducing the volume of configuration space available for hydrogen bonding. Here we show with computer simulation results that this traditional picture, based on average structural features of hydration water, configurational properties of single water molecules, and up to pairwise correlations, is not correct. Analyzing collective fluctuations in water clusters we are able to provide a fundamentally new picture of HE based on pronounced many-body correlations affecting the switching of hydrogen bonds (HBs) between molecules. These correlations emerge as a nonlocal compensation of reduced fluctuations of local electrostatic fields in the presence of an apolar solute. We propose an alternative view which may also be formulated as a maximization principle: The electrostatic noise acting on water molecules is maximized under the constraint that each water molecule on average maintains as many HBs as possible. In the presence of the solute the maximized electrostatic noise is a result of nonlocal fluctuations in the labile HB network giving rise to strong correlations among at least up to four water molecules.