They were 'granted' this at the Conference of Cairo in 1943 by FDR and Churchill. The Americans needed the Chinese to continue the fight against the Japanese on the mainland, since most of the Japanese armed forces were occupied fighting the Chinese United Front (Both the Kwomingtang and the Communists). At Cairo FDR also was already preparing his 'New Order' by trying to get all allied powers to sign off on the United Nations Idea, and by giving China the seat they also participated in the United Nations.

It's also important to note that FDR and Churchill negotiated with Chiang Kai Shek, and as such the veto power in the Security Council went to the Kwomingtang, even after they were forced in exile to Taiwan. Only after the thaw in the '70s did the Pernament seat at the Security council and veto-right go to the Peoples Republic of China, instead of the Republic of China (Taiwan).

Source: KERREMANS, B. and LAENEN R., International Politics since 1945, Leuven, 1999.

Edit: Spelling Errors

FDR's post-war vision is very different than the what the policy Truman implemented after FDR's death. FDR imagined the UN would be policed by the worlds 4 major powers: the US, USSR, England, and China. Together, these 'Four Policemen' would actively prevent conflicts around the world to prevent a future World War. They would lead the UN and give it real power to stop conflict- a direct contrast to the failed League of Nations. FDR saw no interest in SE Asia (Indochina specifically) and thought it belonged in China's realm of influence, and did not support France's effort to regain it's lost colony.

As the other post mentioned, China was not communist at this point in history. However, FDR's post-war vision did not include an idealogical battle against Communism. Lasting peace was his objective, not a Cold War with USSR, which led to comprise (critics would describe it as capitulation) with Russia over the issue of Poland soviergnty and USSR's declaration of war on Japan. So, the fact the China might fall into the hands of communism was not a deal breaker for FDR.

To answer your final question about China's weakness: China had existed as the major power in the region for thousands of years. However, due to European and Japanese imperialism in the last 200 years, China's position had fallen immensely. But it's size, population, and potential for economic growth and military growth made it a realistic choice to become the future power in the region.

Source: His Final Battle: The Last Months of Franklin Roosevelt Book by Joseph Lelyveld

This eventually happened in october 1971 when the US and decrease that of the advisors who were in his book Nehru The Invention of India.