ingratiate yourself with someone through obsequious behaviour.
Curry here means groom a horse or other animal with a coarse brush or comb. The phrase is an early 16th-century alteration of the Middle English curry favel - Favel (or Fauvel) being the name of a chestnut horse in an early 14th-century French romance who epitomized cunning and duplicity. From this to groom Favel came to mean to use on him the cunning which he personified. It is unclear whether the bad reputation of chestnut horses existed before the French romance. But the idea is also found in 15th century German in the phrase den fahlen hengst reiten (ride the chestnut horse) meaning behave deceitfully.