How Taiwan joined China: Before the European colonization there were few Chinese people in Taiwan. There were some in coastal fishing villages, but most of the people in Taiwan lived in tribes/chiefdoms/kingdoms that spoke austronesian languages. In early 1600s first the Spaniards then the Dutch arrived and began colonizing. The Dutch needed more labor and encouraged Chinese people from Fujian (province across the strait) to move to Taiwan to work as farmers and fur trappers.

In the late 1600s the Manchu invaded Ming Dynasty China. An admiral and Ming Dynasty loyalist invaded the Dutch colony (spanish had already left). He drove out the Dutch and established a base to continue fighting the Manchu from. He failed, the Manchu established the Qing dynasty and the Qing seized the Ming holdings in Taiwan. Immigration continued largely from Fujian. Many indigenous sinicized overtime, many others continued their lifestyle.

Japan period: The Japanese seized Taiwan (and other stuff) in late 19th century after defeating Qing dynasty in a war. There was resistance to Japanese which was brutally put down. The Japanese recognized the indigenous tribes, but strictly discouraged use of languages other than Japanese in school and government. Many very old people in Taiwan still know some Japanese because of this period.

KMT: The losing side of the Chinese civil war was the Republic of China (ROC). They were led by the far right pro-west KMT. The ROC fled to Taiwan with many supporters, soldiers, pro KMT refugees etc. There were many people in Taiwan who did not like this, some even supported the communists. Chiang Kai Shek, the president/dictator of the ROC, initiated the white terror where tens of thousands of anti-KMT people were killed and hundreds of thousands of others were imprisoned. Today, there is a lot of criticism of Chiang Kai Shek especially among young people.

The ROC maintained they were actually the legitimate government of China even as most nations, including their allies switched recognition the PRC. In fact, the ROC claims greater borders than the PRC does since the PRC negotiated disputed borders that reduced it's territory. Taiwan passports today still say ROC, the KMT logo is still on the Taiwan flag and many government buildings.

In the 1990s (after the KMT dictatorship ended) the PRC and ROC arrived at the One China Policy. This is also called the "consensus". They both agreed there was only one China even though they disagreed about which government was legitimate. Independent Taiwan is a line in the sand for the PRC, which they say if crossed would lead to war.

By this time there were many people in Taiwan who wanted to be separate from China. This is partly attributable to a hundred years of anti-communist propaganda, but also do to hostility engendered by the KMT's white terror. The US encouraged keeping Taiwan under ROC government because it was useful for Cold War (Taiwan has been referred to as "an unsinkable aircraft carrier").

There are few ways this attitude divides. Young people are much more pro independence, old people are not. The DPP party is pro-independence but would never make the move. The KMT party (who are in opposition currently) are against independence. As are some other conservative elements of society (like the military). Indigenous people in Taiwan tend to be pro-KMT and skeptical of the DPP and hence skeptical of independence. They often see the DPP as more narrowly interested in what's good for Hakka Chinese (the main han group that came from Fujian).

I'm sure I'm missing a lot of nuance, I'm not from Taiwan and I'm pretty much illiterate in Chinese and conversationally I'm incapable of discussing politics.

Legally Taiwan is very clearly part of China. It was illegally seized by the Japanese and then transferred to the control of the losing side of the Chinese civil war. Most of the people in Taiwan are ethnically and culturally Chinese and nearly everyone speaks/writes standard Chinese.

Imo, the only people in Taiwan who might have claim to independence are the indigenous people, but the pro-Independence movement in Taiwan is not about them; it's about Chinese people who now assert Taiwanese identity against Chinese identity. But, again, I'm sure I'm missing nuance.

Edit: presently, situation isn't much different than past. Both sides regularly saber rattle at each other, but many businessmen from Taiwan are heavily economically connected to mainland China. Many retired Taiwan military officers acknowledge they would lose in a conflict with China. Taiwanese independence supporters point to HK as their future if things don't change. Most mainland Chinese unequivocally reject Taiwan independence.