Hermione already knew about magic long before getting her Hogwarts letter, and had been studying old hogwarts texts for years.

This came about because the Grangers live down the street from the Tonks family, and Hermione caught, and used to "blackmail" her, Tonks using her Metamorphagus ability to prank/scold some of Hermione's bullies.

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Tonks told Hermione about magic, about how it has to be kept a secret, and about how she intervened with the bullies because she's seen Hermione do accidental magic before.

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So Hermione grows up with a "big sis" figure in Tonks.