The story
In the sense of a bribe, the phrase is a punning reference to the palm of the hand and also to oil as lubricant rather than foodstuff.
Etymology from Wiktionary, CC BY-SA 4.0.
palm oil
Today it means: An edible plant oil derived from the mesocarp (reddish pulp) of the fruit of the oil palm (Elaeis guineensis). Full definition →