The story
An allusion to a butcher who cheats customers by surreptitiously using a thumb to apply extra pressure to the scale when weighing meat to calculate the price of a sale.
Etymology from Wiktionary, CC BY-SA 4.0.
thumb on the scale
Today it means: An act of bias or a tactic for cheating which creates a situation that unfairly benefits one party involved in an interaction. Full definition →