This answer is full of unmarked spoilers to the manga and the anime. The anime ends maybe halfway through the manga, and further on in the manga it is revealed that Akito is
From the wiki:
After Yuki left the Main House to live with Shigure, Akito's mother Ren commented that the cursed Sohmas were abandoning Akito, just as Ren always predicted they would. Akito denied this and claimed they would all come back to her because of their inseparable Zodiac bonds. She and Ren made a dare in which Akito was to allow the cursed Sohmas to form bonds with "outsiders" to test whether they would return to Akito's side. If they did not return, Akito would have to bow before Ren then leave the Sohma family.
Ren is not fully sane. Her obsession with Akira and jealousy of Akito's relationship to him caused her to emotionally abuse Akito. There's a whole backstory on why Ren is so abusive, but I don't think it's entirely relevant. Due to Ren's emotional abuse focusing primarily on Akito being unloved, Akito's deepest fear is that she ends up alone, with everyone having left her. She recognises the power, as God, she has over the cursed Sohmas so she abuses that to ensure they never break the curse and presumably leave her. When Kureno, the Rooster, breaks free of his curse (about ten years before the events of Fruits Basket), Akito confines him to the Main House for fear of him abandoning her. She even goes so far as to attempt to keep him there through seduction, and although Kureno does not fall in love with her, he does have sex with her. At the same time, however, Akito pursued a relationship with Shigure, whom she truely loves. Shigure recipricated those feelings, but Ren intervened and convinced Shigure to sleep with Ren in revenge for Akito's sex with Kureno. When Akito discovered this, she exiled Shigure from the Main House.
Years later, Hatsuharu convinces Shigure to allow Yuki to stay at his house due to the extreme emotional abuse Akito inflicts on Yuki. Kyo also winds up staying at Shigure's as he is forbidden from staying in the Main House. Ren continues to torture Akito by declaring that all three's departure suggests that they are abandoning her, and, since Ren sucessfully convinced her that, given the choice, the cursed Sohmas would rather abandon Akito than support her, Akito pitifully clings to the belief that the Zodiac bonds will force them to return. In defiance of her mother, Akito allows Tohru to stay to prove Ren wrong, and reaffirm Akito's conviction that the Zodiac bonds trump all.