Why does the Förster resonance energy transfer scale as 1/r⁶?
From this article by Förster himself:


  This coupling is strongest if the corresponding optical transitions in
  both molecules are allowed for electric dipole radiation. Then these
  transitions are coupled not only to the radiation field but also to
  each other. Naturally, the interaction energy is of a dipole-dipole
  nature, depending on an inverse proportionality to the third power of
  the molecular distance. The probability of energy transfer is then
  proportional to the square of this interaction energy and decreases,
  therefore, with the sixth power of the distance.


This paper does not give a detailed derivation though. But in his seminal paper from 1948 Förster goes through the derivation in detail (unfortunately it is not in English but in German and the condition of the text is not very good).