What was the purpose of the "tail prop" on the Ilyushin Il-62 and why was it necessary?
Are you asking about the rear support wheel? It is for supporting the aircraft and avoiding tilt on the ground.

Image from wikimedia commons. Autor:M Radzi Desa

As the aircraft had four engines in the aft, the cg would've been aft too; so they added a tail prop to prevent tilt on ground.

Interestingly, this doesn't seen to be unique for Il-62. Another aircraft with rear mounted engine, Sud Aviation Caravelle, has something similar to this. 

Image from wikimedia commons. Author: Eduard Marmet