So to be clear the Republic of China was not barred from being in the UN, it was in fact a founding member, original P5 member of the Security Council, and ally from WW2. However that government then managed to lose its civil war with the Communists under Mao and was pushed off the mainland to the island of Taiwan.

That government then proceeded to continue to hold the China seat at the UN. The People's Republic of China then on the mainland was seen as another large country which had fallen to communism and a threat that must be countered, especially so close to allies such as Japan and South Korea(with outright conflict threatening following their intervention in the Korean War). Under the emerging policy theories of Containment and the so called Domino theory, Communism should be met force for force in kind wherever it sought to expand or assert itself. By doing so it could prevent a takeover of an initial country which would then increase the chances of neighboring nations falling as well, in particular as it related to SE Asia.

However the Communist Bloc was not a uniform lockstep creature, with relations between the USSR and PRC often very frosty. This was something that Nixon in the early 70's sought to exploit and accentuate. Thus began a whole series of deals and negotiations which in the end began the economic relationship between the nations, restoring of diplomatic relations, recognizing the PRC under the "one China" policy, and going along with votes in the UN to remove the China seat from Taiwan and allow the mainland government to fill the spot.