Well, if you compare China's (the PRC) timeline with that of the USA and USSR, you will see that it was simply not on their to do list - they had other problems to deal with.

The USSR and USA had both just emerged from WW2, victorious and predominant world powers, each controlling half of Europe with their allies alongside them, along with newly developed technology and captured Nazi scientists. Since they were both now at peace, despite the Cold War, they were both free to develop tech for space travel, satellites, and the like. Space was seen as the quickest route to intimidate the other side, which I'm sure you've heard, and so they both spent much of the 50s and 60s developing space technology.

Now let's look at China. While victorious against the Japanese in WW2, the Chinese immediately went back to fighting amongst themselves - namely, the Nationalist led by Chiang Kai-Shek vs. the Communists led by Mao Zedong. After 4 more bloody years of warfare, the Communists win and establish the PRC on the mainland while the nationalists fled to Taiwan, where they remain. Mao Zedong now had to figure out how to catch up with the rest of the world (China was not a very advanced state and had been ravaged by years of warfare with the Japanese and amongst themselves).

Mao initiated the Great Leap Forward in the 50s to attempt quick industrialization. It failed, miserably, and millions died. In danger of losing his grip on power, Mao initiated the Cultural Revolution in the 60s to get rid of whom he saw as a rival, where many more died or were imprisoned. During this time, China and the USSR also went from being allies to enemies, since Stalin was dead and Khrushchev's policies weren't popular with Mao and his allies. So, that sapped China of any technological assistance from their former friends. After Mao died in 1976, which somewhat marked the end of the Cultural Revolution, there was a struggle for power. After some trouble, Mao's former protege Deng Xiaoping came into power and brought some stability and liberalization to the economy, turning China into what it is today - an economic powerhouse.

With Deng, China was now able to focus on expensive projects like space exploration. In short, China was undergoing serious problems at home and couldn't afford to spend much time or money focusing on space and rockets.