Highly conductive solid electrolytes (SE) that are stable in contact with Li and cathodes are required for all solid state battery or high energy Li-S and Li-air batteries. With the advantages of high conductivity (>10^-4 S/cm) and excellent stability, Li-stuffed garnet Li₇La₃Zr₂O₁₂ has emerged as one of most promising SEs for Li batteries. Even though the Li ionic dynamics in garnet have been studied extensively, the couplings between Li transport and the vibrations of framework atoms, namely phonons, have seldom been studied, and the controversies in Li transport mechanism and superionic transition have not been settled. In this study, we synthesized phase-pure tetragonal and cubic Li₇La₃Zr₂O₁₂ using a solid state reaction method, and systematically studied their thermal conduction, lattice dynamics, and phase transition. Cubic phase, stabilized by 0.3 mol aluminum, shows much lower thermal conductivity k than that of tetragonal phase, e. g. 1.7 vs. 4.3 W/m-K at 70 K and 1.4 vs. 2.4 W/m-K at 300 K. More specifically, k of cubic phase is almost temperature independent and approaches the amorphous limit, indicating very strong phonon scattering; while the tetragonal phase shows a decreased k with increasing temperature, typical of crystalline solids with the Umklapp processes dominating phonon scatterings. By examining the lattice dynamics and velocity auto–correlation function (VAF) in the cubic phase, we found a high frequency hopping mechanism of Li rather than a liquid-like diffusion mechanism, and Li atoms predominantly contribute to the low energy lattice excitations. This indicates that Li atoms ‘rattle’ intensively in their equilibrium position before hopping to another sites, and the ‘rattling-like’ mode strongly correlate with lattice phonons to block their transport. Moreover, the Li-framework couplings also cause the instability of Li atoms in their positions and promote the hopping responsible for the high ionic conductivity. Our study provides additional understanding of ionic transport mechanism in garnet and insights into thermal management of all solid state batteries.