Hybrid organic-inorganic halide perovskites, particularly the methylammonium lead trihalide perovskites (MAPbX3; MA=CH3NH3+ , X=Br− or I−), have gained extensive interest for optoelectronic, photovoltaic and thermoelectric applications recently. In the hybrid perovskites, the A-site species of the conventional inorganic perovskite, such as Cs+ in CsPbI3, is replaced by an organic molecule (e.g. CH3NH3+). The introduction of a molecular species reduces the sublattice symmetry and introduces local internal vibrational modes. Recently the dynamics of methylammonium ions have been recognized to be of importance for the thermal and electrical properties, but the detailed structure-property-process relationships are still lacking. In this work, we combine ab initio lattice dynamics and molecular dynamics simulations to connect both the harmonic spectra and anharmonic interactions of vibrational modes, as well as the correlated disorder between the molecular orientation and the vibrational modes of the underlying octahedral network, to the phase transition and thermal properties of bulk MAPbI3. By comparing to inorganic perovskites (CsPbI3), we first show that the ultra-low lattice conductivity currently under debate arises primarily from the strong anharmonicity, although the low-group velocity cannot be neglected. Based on the resolved interactions between individual modes, we show that the internal modes of the organic molecules interact strongly with the soft modes residing on the inorganic framework at the M and R high-symmetric points, which facilitates the phase transitions between orthorhombic, tetragonal and cubic phases. Furthermore, spectral energy density analysis demonstrates the spectral broadening of vibrational modes due to the presence of orientational disorder. This study will also be of interest to the thermal stability and moisture resistance of hybrid perovskites.