In a speech to the eighth meeting of the central peoples government counsel on June 28, 1950, Mao said "The Chinese people had long ago declared that the affairs of the various countries throughout the world should be run by the peoples of the respective countries themselves, and the affairs of these should be run by the people of Asia themselves and not by the United States. United States' aggression In Asia will only arouse the extensive and resolute resistance of the peoples of Asia."
Mao and other Communist Party members talked much of making world peace. They also talked much of Taiwan, and taking it away from the grasp of the United States. How could Mao and other Chinese officials justify the taking of Taiwan, when the people of Taiwan wanted to self govern themselves, and that such aggression would certainly cause war?
The People's Republic of China is the product of a civil war between the Communists, and the Nationalists who were the Chinese national government from 1911-49. When the Nationalists lost, they retreated to Taiwan, and to Nationalist leader Chang Kai-shek and his generation in Taiwan, the Nationalist government continued to be the only legitimate government for all of China. To Mao and his generation on the mainland, however, Taiwan was a renegade province that had yet to be assimilated into the People's Republic. Neither side saw Taiwan as an independent nation, but as a part of China. It's only been recently that the people of Taiwan have started talking openly about independence.
The U.S. backed the Nationalists in the civil war, and the U.S. military is the primary reason Taiwan was never assimilated, which the Communist government saw as an interference in the internal affairs of China - hence the speech you cite. Part of the agreement between Nixon and Mao that normalized relations between the U.S. and China in 1972, however, was that Taiwan is never to be referred to an an independent nation, but always as an integral part of China.