One somewhat simplified way of looking at it is that the London dispersion forces are a dipole-dipole interactions. The interaction between two dipoles depends upon the relative orientation of the two dipoles. Some positions are attractive and some are replusive, but averaged over all the positions (and considering that lower energy positions are favored by Boltzmann statistics), the interaction is attractive with potential energy having an 1/$r^6$ dependence.
See for example Stephan Franzen's lecture
and especially the corporate Intermolecular Van Der Waals page
as well as Fritz London's The General Theory of Molecular Forces Trans. Faraday Soc., 1937, vol. 33, pages 8-26.
for more on the 1/$r^6$ dependence of dipoles.
However, a proper quantum mechanical treatment shows that London dispersion interaction is not always proportional to 1/$r^6$
For example, between two hydrogen atoms in 1s states, the potential energy is proportional to 1/$r^6$ but
between a hydrogen atom in a 1s state and a hydrogen atom in a 2p state, energy is proportional to 1/$r^3$
See: Complement $C_{XI}$ of Quantum Mechanics vol. 2 by Cohen-Tannoudji et al.
Also, in 1948 Verwey and Overbeek demonstrated experimentally that the London dispersion interaction is even weaker than 1/$r^6$ at long distance (say hundreds of Angstroms or more).
Casimir and Polder soon thereafter explained with quantum electrodynamics (QED) that the dependence should be 1/$r^7$ at relatively long distances.
For a nice historical overview and QED prespective see Some QED vacuum effects: van der waals forces in The Quantum Vacuum by Peter W. Milonni.