A basic, phenomenological understanding of the transition dipole moment can be obtained by analogy with a classical dipole.  While the comparison can be very useful, care must be taken to ensure that one does not fall into the trap of assuming they are the same.
In the case of two classical point charges,  and , with a displacement vector, , pointing from the negative charge to the positive charge, the electric dipole moment is given by
In the presence of an electric field, such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net torque on the dipole.  The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole:
Similarly, the coupling between an electromagnetic wave and an atomic transition with transition dipole moment  depends on the charge distribution within the atom, the strength of the electric field, and the relative polarizations of the field and the transition.  In addition, the transition dipole moment depends on the geometries and relative phases of the initial and final states.