Final answer:
The royal family tried to escape from Paris to the Austrian Netherlands in June 1791 but were caught and brought back, leading to a decline in their support and the rise of radicalism in the French Revolution.
Step-by-step explanation:
When the royal family tried to escape from Paris, they were attempting to flee France and head east towards the Austrian Netherlands (present-day Belgium). Their aim was to reach a safe haven from where they could possibly gather support to counter the revolution that was unfolding in France. King Louis XVI and his family attempted to escape in June 1791, but their plan failed. They were recognized in the town of Varennes and were arrested and brought back to the Tuileries Palace in Paris, where they were placed under house arrest. This event, often referred to as the Flight to Varennes, was a turning point in the French Revolution, as it significantly eroded the king's credibility and trust among the French people. The subsequent events led to the rise in radicalism, the establishment of a republic, and ultimately, the execution of Louis XVI and his wife Marie Antoinette.