From Ryan's explanation, "burned at the stake" is used figuratively in modern English to mean the person faced a serious (and on occasion, perhaps unfair) reaction from other people. "The member of parliament was burned at the stake by the press for his comments about lower welfare for single mothers." (OK, not literally, but it's a pleasant image.)

Yes, it was the method of execution used by the Catholic Church against "heretics." The method was used because the clergy were forbidden from "spilling blood." ( Presumably they thought that broiling blood was just fine.) Joan of Arc was burned at the stake. The person is tied to a pole, standing on a big pile of wood. They set the wood on fire.

death by burning after a criminal is tied to a long pole.