Based upon his position as a member of the last legitimate French government, the British recognized de Gaulle as leader of the "Free French" assisting his plan for resistance to the pro-German "Vichy" government in France. The British also recognized governments-in-exile for other countries occupied overrun by the Germans, e.g., the Free Polish, Free Czechs, Free Norwegians, etc. These were of value for propaganda purposes, as sources of spies and saboteurs to be sent back to the occupied countries and for helping thousands of young men to escape from Europe and form regular military units to fight against the Germans.
Unfortunately for de Gaulle, the British, fearing that the French fleet at Mers-el-Kebir, (in North Africa) was about to surrender to the Nazis, tried to force it to defect to him. When the French refused, the British attacked, sinking the ships and de Gaulle's hope of widespread support in France.
Another problem for de Gaulle was that the United States had already recognized the "Vichy" government and Roosevelt felt he could not legally recognize the Free French as well. When America entered the war, this led to a confusing "Two France Policy", with the British and Free French being at war with the Vichy government while the Americans were not. Even after U.S. forces invaded French North Africa in November, 1942, FDR wanted to replace de Gaulle with his rival, Gen. Giraud, which led to endless friction in the Allied high command.
However, by 1942, in spite of Mers-el-Kebir, the brutality of life under the Nazis had convinced most Frenchmen that de Gaulle had been right to reject the surrender. Free French forces grew stronger, and resistance to Vichy rule increased. These forces played a vital role in the liberation of France and the final defeat of Nazi Germany.