Read the extra summary first prolly

To keep it short, the AI of Sakura gains true consciousness, something that shouldn't happen, and falls in love with MC. To avoid deletion, she creates a backup, BB, who gains self-awareness and takes control of the Moon Cell, trapping a handful of Masters on the Reverse Side of the Moon, around the time of the Fourth Round. There, MC, Leo, Julius, Gatou, later Rin and Rani, and a mysterious woman known as Kiara with her Servant Andersen cooperate to reach BB past the Sakura Labyrinth and escape. The story goes slightly differently if you choose Normal route or True route then. In Normal route you fight all bossess, including BB's Alter Egos Passionlip and Meltlilith, until BB loses control and hacks the Moon Cell Core, planning to use it to wipe out humanity. You defeat her, and return back for the normal Extra story. In True Route instead, you spare the Alter Egos after battling them, though they're killed by someone. When you get to BB, she is killed as well by Kiara, who reveals herself as the true antagonist and the one who gave Sakura and BB self-awareness to begin with. She then uses Andersen's Marchen Meines Lebes to become a being close to a Beast in nature, you defeat her, the Core then tries to erase everyone but your Servant sacrifices itself to protect you (except Gil, he is too badass to sacrifice himself, he's the only one that survives. That's why most people use Gil when doing True Route). Sakura then uses the Core to create a physical body for MC and herself, and you leave the MoonCell. Note that the digital MC actually still exists now, and will still follow Extra story, so Extella is still able to happen