We investigate the role of rattling guest atoms on the lattice thermal conductivity of a type-I clathrate Ba_{8}Ga_{16}Ge_{30} by first-principles lattice dynamics. Comparing phonon properties of filled and empty clathrates, we show that rattlers cause tenfold reductions in the relaxation time of phonons by increasing the phonon-phonon scattering probability. Contrary to the resonant scattering scenario, the reduction in the relaxation time occurs in a wide frequency range, which is crucial for explaining the unusually low thermal conductivities of clathrates. We also find that the impact of rattlers on the group velocity of phonons is secondary because the flattening of phonon dispersion occurs only in a limited phase space in the Brillouin zone.