Phonon transmission across an interface between dissimilar crystalline solids is calculated using molecular dynamics simulations with interatomic force constants obtained from first principles. The results reveal that although inelastic phonon-transmission right at the geometrical interface can become far greater than the elastic one, its contribution to thermal boundary conductance (TBC) is severely limited by the transition regions, where local phonon states at the interface recover the bulk state over a finite thickness. This suggests TBC can be increased by enhancing phonon equilibration in the transition region for instance by phonon scattering, which is demonstrated by increasing the lattice anharmonicity.