To better understand the ionic conduction in lithium garnet oxides, we employed molecular dynamics simulation to investigate the local structure and dynamics of a model material Li7La3Zr2O12 and origin of its tetragonal to cubic phase transition. Our simulations were able to produce lattice parameter, neutron scattering, and conductivity data close to those gathered using experimental techniques, which allows us to study atomic-scale details of this complex material. First, it was found that lithium atoms primarily perform oscillation and "structured diffusion" dynamics in the tetragonal and cubic phase, respectively. Second, we believe that the tetragonal to cubic phase transition is an entropy-driven one that involves redistribution of lithium atoms among all tetrahedral sites. The transition is likely to initiate on the tetrahedral 8a site but needs the cooperation of neighboring octahedral 32g Li as relay atoms. Finally, it was found that a few types of lithium clusters dominate in both phases of Li7La3Zr2O12, which leads to highly correlated motion of lithium atoms. The local symmetry of these clusters dictates a "center-pass" mechanism as lithium goes through the bottleneck.