Time-resolved fluorescence spectroscopy is used increasingly to probe molecular motions at the aqueous interfaces of biological macromolecules and membranes. By recording the time variation of the fluorescence frequency, thermal atomic fluctuations in the vicinity of the chromophore can be probed. From such fluorescence Stokes shift (FSS) experiments, it has been inferred that water motions in the hydration layer are slowed down by 1-3 orders of magnitude. To provide a more secure foundation for the interpretation of FSS data, we use molecular dynamics simulations to examine the molecular origin of the FSS from a tryptophan residue in a protein. By using linear response theory to decompose the FSS into its water and protein components, we find that the water component dominates the static FSS but decays rapidly. Thus, after a few picoseconds, the FSS essentially reflects protein dynamics, including the self-motion of the chromophore. Because of its collective nature, the FSS response is insensitive to the motion of individual water molecules. Collective water displacement by slowly fluctuating protein groups introduces a long-time tail in the water autocorrelation function, but this dynamic coupling is hardly manifested in the observed FSS. Our analysis reconciles FSS data with the picture of a highly dynamic hydration layer, derived mainly from magnetic relaxation dispersion and simulation studies, and calls for a revision of previous interpretations of FSS decays in terms of slow hydration dynamics at biomolecular and other interfaces.