Why and how did the CIA/OSS support non-communist French trade unions after the WWII?
Trade unions have a certain influence over the economy and politics in each country and because they're organized in international organizations, their influence has a geopolitical dimension, too.
After 1945, the Soviets and the communists had a lot of political capital due to the role of the USSR role in defeating Nazism.
The Soviet Union was basically capable of replacing the (non-communist) International Federation of Trade Unions (which existed in 1919-1945 and had headquarters in Amsterdam) by the (pro-communist) World Federation of Trade Unions (whose headquarters were in Prague; after the fall of communism, the importance of this entity declined when communism ended and as a result, in 2006, the headquarters were moved from Prague to Athens).
The Soviets and their sympathizers were able to promote their alternative views about the Marshall Plan and other things. This was dangerous from the American perspective. The U.S. and British trade unions would generally oppose the Soviet-led international organizations but the continental West European trade unions were somewhere in the middle.
France was the ultimate "swing state" and CGC and Force Ouvriere were the French non-communist trade unions who opposed the Soviet line in the international organizations (the French CGT was a pro-communist example). That's why they got some funding from the U.S.
At the end, in 1949, the (non-communist) International Confederation of Free Trade Unions (led mostly by the Americans and the British) separated from the Prague-based pro-communist international umbrella organization.
However, the labor movement in Europe became so polarized that the entities were largely considered appendices of their political sponsors and the unions' importance decreased as a result.