To the first order it is the fact that a liquid only propagates pressure waves while a solid can propagate shear and pressure waves (which propagate a different speeds).
Pressure waves go through the liquid part, hit the solid part, some of their energy is converted into shear waves which go through the solid core, convert back to pressure waves on the other side, and eventually hit the surface again. By measuring propagation time, direction, etc, the full picture is only compatible with the pressure / shear / pressure double conversion.
