To seek to get into someone's good books; to ingratiate oneself.
Curry refers to a horse riding term for grooming or rubbing down an animal. The favour is an alteration of the word Favel. Favel was the name of the half horse, half man Centaur in the early 14th century French satirical romance Le Roman de Fauvel. This beast was cunning and evil and it was just as well to keep on the right side of him. To curry him kept him in a good mood.