Before answering, just to take issue with the premise of the question. Yes, France received a permanent seat on the Security Council, not to mention her own occupation zones in Germany and Austria. But France was not accorded a status anything like the "Big Three" in most other respects. From May 1943 De Gaulle was operating independently from French territory in Algiers but he was consistently slighted and ignored by the British and Americans. He was especially wounded by the practice of the American authorities coming and going as they pleased in Algeria without even a nod to French sovereignty and, later, when he was left in the dark about D-Day. The French were either excluded or allowed only minor roles at the Casablanca, Yalta and Potsdam conferences.

However it shouldn't be surprising that in certain respects (Security Council, occupation zone in Germany etc) France's status as a nation of the first rank was soon restored.

For one thing, De Gaulle, impressively and consistently, had worked for almost nothing else but the preservation and rebuilding of the status of France since founding the Free French in 1940. He certainly valued that task higher than shortening the war or aiding his allies which was why he so maddened the Americans and British. But it did earn their grudging admiration (or at least Churchill's) and, crucially, it meant that no one in 1945 could doubt that France was an independent power and not merely liberated territory with a government installed and directed by the Americans.

Secondly De Gaulle had diligently courted Stalin between 1943 and 1945 and, while they had nothing like a friendship or alliance, it meant that Moscow wasn't unwilling to accommodate the French at the top table (especially if they could be used against the Americans, see above).

And also remember that the statesmen reshaping the world after 1945 weren't operating in a historical vacuum. Their perceptions of the relative significance of nations would have been altered, certainly, but not wholly revised. France after all had had one of the largest armies and air forces in the world in 1939, and the fourth largest navy.

Even after the war France remained second only to Britain in terms of imperial possessions. Her soft power (though in decline) remained substantial throughout the 20th century. The role and importance of the French language throughout Europe shouldn't be underestimated.

Finally it would be a mistake not to recognise that in 1945 the British and Americans needed a friendly continental land power that to take a lead in policing post-war Europe and the only conceivable candidate was France. Britain didn't see herself as a continental power and - prior to the Marshall Plan and Truman Doctrine - there was no guarantee the Americans were going to hang around.