A high-pressure study of the structural and thermal transport properties is carried out on one of the most efficient filled skutterudites, ${\mathrm{Yb}}_{0.3}{\mathrm{Co}}_{4}{\mathrm{Sb}}_{12}$, to understand the relatively low thermal conductivity behavior in this family. By combining x-ray diffraction and Raman scattering measurements, we detect a phase transition at around 12.4 GPa. The mode Gr\"uneisen parameters of the observed phonon modes are obtained from the determined bulk modulus and the phonon frequency shifts with pressure. The strong anharmonicity in this material is demonstrated by the obtained large average Gr\"uneisen parameter. We also find the depressed group velocity within the low-frequency range, the flat guest mode avoided crossing with the acoustic-phonon mode, and the significant contribution of optical phonons. The hybridization of the guest atom and host lattice and the related enhanced anharmonicity are suggested to account for the low lattice thermal conductivity in the studied system. These findings provide new insight into how phonon-phonon interactions lower lattice thermal conductivity in this important thermoelectric family.