During the Chinese Communist Revolution the previous government of China retreated to Taiwan. The official name of the Taiwanese government is “Republic of China”, demonstrating that their legitimacy stretches back to this time period.
While Chinese settlers in Taiwan dates back further, China claimed the island in the 17th century. The island was under Japanese rule for about 50 years, before being returned to China in the post WW2 treaties.
The history is deep, and Taiwanese independence is a relatively new thing.
It is based on a manifest destiny ideology that the PRC is the only legitimate heir of Chinese culture and history, and sole rightful ruler of all people of Chinese descent.
From a security standpoint it would be like if a group of US politicians and other elites took over Cuba and called themselves The Republic of The United States. Yeah, an island nation wouldn’t win a fight, but their intentions are in the title.
ROC is not going to conquer or invade PRC. The is ideological and presented by ROC merely existing.