When can I cast Eternal Scourge from exile?
You can only cast Eternal Scourge from exile during your pre-combat or post-combat main phases if the stack is empty. (eg. with the same timing restrictions as casting it normally)

This is addressed specifically in Eternal Scourge's rulings:


  7/13/2016: You must follow the normal timing permissions and restrictions and pay its cost to cast Eternal Scourge from exile.


As for what "normal timing permissions and restrictions" means, look to the comprehensive rules for casting a creature spell:


  302.1. A player who has priority may cast a creature card from his or her hand during a main phase of his or her turn when the stack is empty. Casting a creature as a spell uses the stack. (See rule 601, “Casting Spells.”)


The ability text on Eternal Scourge just lets you substitute "from his or her hand" with "from exile", but nothing else about casting creatures changes.

As a side note, if you have any spells that would modify when you are able to cast creature spells (for example, Aluren) unless they specify "cast from your hand" those modifications would apply to Eternal Scourge as well.