Queen Anne was the daughter of James II of England (a Stuart), who was deposed as king by the "Glorious Revolution" of 1688 that brought his daughter Mary, and her husband, William of Orange, to the throne as William and Mary. James had been Catholic. William and Mary were Protestant. After Mary and then William died, Mary's sister, Anne, was named Queen. She, too, was a Protestant, but it was known that James' son and Anne's brother, a/k/a the Pretender, had supporters who wished for the restoration of the Stuarts and possibly Catholicism.
In 1714, Queen Anne was known to be in ill-health. She had no heirs because her children had all died in childhood. Her official successor was to be her nephew, the son of her sister Sophia, who had married a German potentate. That nephew, George of Hanover, was German, not British. This state of affairs had the country on edge, as it was anticipated that, upon Queen Anne's death, a civil war could break out between the factions of the Stuarts and the Hanovers.
When Queen Anne finally died, her death was not announced until it was clear that George of Hanover would become George I of Great Britain and that there would be no war. By the time of the official announcement of the Queen's death to the public, everybody who mattered already knew that she had died.
And so "Tell 'em Queen Anne's dead" became the equivalent of the modern, "tell me something I don't already know."