The minds of people are now accustomed to regulation, and civilization has become stagnant. There are no great tragedies, but neither are there new discoveries. Pessimism reigns supreme in this insipid world, where we can only bide our time until death. This is our reality."
The world of Extella differs from that of Fate/stay night in that it follows an alternate history, with events having started to diverge in 1970. Fate/EXTRA and its sequels are set on Earth and the Moon, starting in 2030.
Fate/EXTELLA takes place after the final battle in Fate/EXTRA. The Main Character, with the help of main Servant Nero and sub-Servant Tamamo no Mae, emerged victorious from the Holy Grail War, and advanced to the Moon Cell core.
However, Tamamo became lost on the path to the core, thanks to certain obstacles, and only Nero and the Main Character arrived to face off against their final foe, Twice E. Pierceman.
After his defeat, the Main Character entered the Moon Cell core and interfaced with it, invoking the "wish-granting" power of the Holy Grail and altering the state of the virtual world known as SE.RA.PH.
That alteration not only changed the rules, so that defeated participants and AIs were not simply deleted, but it changed the determination of the Moon Cell as well. The Main Character wished to end the vicious cycle wherein those who fulfilled their roles were removed and reformatted to serve in the next War. Instead, per the Main Character's wish, those who have completed their duty can now seek new purpose, living for their own reasons. This critical change allowed the Main Character to go on living as well.
In accordance with the new settings, the Moon Cell began to weave together the individual sections of SE.RA.PH and created a new land. That was the birth of a heavenly body, growing and thriving in virtual space alone. The Main Character was sent back to SE.RA.PH, and as the sole human in the new virtual world, found new expectations and responsibilities waiting…