There's more to the story imo.

What everyone else said is true. The nationalists and communists waged competing revolutions to overthrow the Qing and the warlords and the Japanese Occupation. The communists won, establishing the PRC on the mainland while the nationalists lost, establishing the RoC on Taiwan. Both claimed control over both the Mainland and Taiwan.

The US propped up the RoC, but also wanted to do trade with the PRC with the goal of putting a wedge between the PRC and USSR as well as encouraging the PRC to liberalize. The PRC would not trade with the US while they recognized the RoC as China, so the US adopted a position of recognizing the PRC as China and the ROC as Taiwan.

They have also, through the CIA, funded cultural projects in order to foster a Taiwanese national identity. This has been incredibly successful to the point where much of the Taiwanese "left" support Taiwanese independence. It also lets the US challenge the PRC's claims to the island by saying "look, the progressives in Taiwan don't want to be part of the PRC, they want to be independent!

So now the US can be hawkish towards China, not under the aggressive posture of claiming the RoC has claim to the mainland, but under the defensive posture of "protecting Taiwanese independence."