The derivation of ETo had developed over many years with several different equations. The latest and hopefully last variant is derived from the Penman-Montieth equation.
Extensive documentation on the methods and concepts can be found in the UN-FAO 56 paper [1]
The derivation for the Hargreaves equation can also be found in the UN-FAO 56 paper.
The History and Evaluation of Hargreaves Evapotranspiration Equation [2] is a more detailed description and background of the Hargreaves method.