I can't specifically answer the question as to "pay". I have it in quotation marks because what a person was allowed to buy in the wartime economy of the period was extremely limited. Arrangements for each country (and perhaps separately each French colony that contributed troops) might be different.

For equipment and rations however, the answer is much simpler: American Lend Lease. American industrial output and food supply underwrote much of the western Allies war effort. French and Polish troops from 1942-45 used primarily American equipment, tanks, artillery, firearms, clothing, helmets etc. Before 1942 this was provided by the British, (themselves backed by a more limited version of the lend lease program before full American entry into the war.) Rations, shipping and all other logistical support were also provided by the common allied supply chain dominated by lend lease material.

This was by no means a matter of charity. By supplying the troops of other nations, America was able to expand it's manpower available to the war effort and reduce American casualties. French and Polish troops more than pulled their weight in some of the hardest fighting in the western theatre.