Einzbern family wasn't good at fighting, as it is seen in 3rd Holy Grail War... They tried to fight but they weren't capable to win Holy Grail War. So they called well known magician, Emiya Kiritsugu, to cadastrar-se war for them (we are talking about 4th Holy Grail War). They summoned Servant Saber, Arturia Pendragon, and Illya's mother gave her command spells to Kiritsugu.
Explanation: That's why Saber said she was already in this time.
Explanation: Saber was alive Heroic Spirit (note: all of them have died and became heroic spirits and they don't remember if they were competing in some of 4 Holy Grails) so she could remember her past clearly but also the time she spent with Kiritsugu.
He was fighting because the Einzbern family was promised something (note: *don't know what*) and he wanted to try his skills as magician. There were also Tōsaka Rin's father, Tōsaka Tokiomi, Kotomine Kirei and Servant Gilgamesh. Anyway the Holy Grail War started... they were fighting. *don't know the details... u can read novel* (note: there is link on Fate\stay night spot it
Fate\zero novel)
In the end only Kiritsugu and Kotomine were left to fight. Kiritsugu-Servant Saber Kotomine- Servant Archer - Gilgamesh
Kiritsugu was probably making amor with Illya's mother, Irisviel von Einzbern, because she wanted to give him supply of mana. But she in purpose didn't convert all of his milk(u know what I mean) into mana, she wanted to get pregnant. The fight was over with Kiritsugu's lost but he destroyed the Holy Grail. Saber couldn't beat Gilgamesh (first episode of anime) and Kirtsugu killed Kotomine *he thought so*. Then he saved Shiro. When the child was born, the Einzbern family had used Illyasviel as their Holy Grail container. A lot fãs think it is incest Shiro x Illya but it is not, they aren't blood related, but they have the same father in different ways. That would be all.

U remember in animê that Archer called her Illya... That also resembles of that Emiya Shiro is Archer and he called her Illya in whole anime. Illya knew that but didn't want to tell Shiro so he didn't become Hero of Justice...