An oscillating dipole emits electromagnetic waves because it contains oscillating electric currents, and because the changing positions of the positive and negative charges make it harder for their electric effects to cancel.
It is easiest to speak in terms of the electromagnetic scalar and vector potentials.
For simplicity, model the molecular dipole as a pair of opposite point charges, separated by a short distance. (Remember, the dipole really comes from the separation between their average positions; but idealizing the molecule as a pair of point charges doesn't hurt the analysis, as long as the molecule is small.)
If the dipole is not changing, then at large distances, the scalar potential due to one end of the dipole and the scalar potential due to the other end will tend to cancel each other out, since the distance to the two charges is almost the same. So the scalar potential will fall off faster with distance than it does for a single charge.
But the news about the charge only travels at the speed of light! If we are slightly closer to one end of the dipole than to the other, then the potential here depends on the charge at the near end of the dipole at some previous time, and the charge at the far end of the dipole a short time before that. So if the charges are moving back and forth at a high speed, the cancellation between the ends of the dipole will be less complete. For instance, the scalar potential here could depend on the charge at the near end at a time when it was positive, but the charge at the far end at a time when the negative charge had not yet gotten all the way there.
If the dipole is much smaller than the wavelength of the light (and air molecules are thousands of times smaller than the wavelengths of visible light), the cancellation becomes linearly less complete as the frequency of the oscillation increases. So at large distances, where the scalar potential of the static dipole would be negligible, the scalar potential due to an oscillating dipole goes up linearly with the frequency.
How about the vector potential? That's easier to figure out. It also varies linearly with frequency, because it's proportional to the current-- and the faster the charges are moving, the more current there is.
The potentials that are produced reverse direction as the dipole reverses direction. If the dipole wiggles back and forth, then oscillating waves of potentials move out from the dipole at the speed of light, with a strength proportional to the frequency of the wiggle. The higher the frequency, the shorter the waves, because they have less time to get out of the way before the dipole changes direction.