In short, the answer is that the Mongol khagan who founded the Yuan Dynasty, Khubilai, deliberately set up his government and state in a Chinese dynastic fashion. He established a regularized government, deployed Chinese-style law codes, surrounded himself with a variety of religious and bureaucratic experts, patronized Confucian schools, reintroduced Chinese monarchical ritual practices, constructed the ritual buildings for those practices in his capital at Dadu, implemented agricultural policies, and attempted to regularize the ad-hoc tax-collection methods that the Mongols had used previously. He and his successors at least wished the Yuan to appear like a Chinese dynasty, and in many respects it was, ritually and functionally.

This isn't to say that the Yuan state copied the old imperial practices of the Tang or Han wholesale. Most notably, the Yuan failed to reintroduce Confucian meritocratic examinations until the 1310s, and deliberately sidelined classically trained Chinese literati by limiting or preventing their ability to hold offices. Southern Chinese (i.e. those who had served the Southern Song) in particular chafed under discriminatory practices intended to prevent Chinese scholars from monopolizing state positions of nearly all levels.

The important thing to note is that ethnicity as we know it today was a very different thing back then. It's difficult to know when and how the people we call Chinese began to think of non-Chinese as ethnically, rather than civilizationally different, but I personally have my doubts that such a transformation occurred before the Mongol conquest. The Mongols were also able to claim the Mandate of Heaven and the "right to rule" by virtue of the simple fact that they had united a huge part of the Eurasian world and expanded the borders of the dynastic state beyond the boundaries of even the great Tang Dynasty.

EDIT: One more thing: the court scholars of the dynasty that followed the Yuan, the Ming, wrote the Yuanshi (History of the Yuan) very soon after the establishment of the Ming Dynasty, indicating that they thought of their Chinese-founded state as the successor of the Mongol-founded one, and legitimizing the Yuan as a proper dynastic state. The Ming founder and first ruler, the Hongwu Emperor, is also recorded in Ming court documents as thinking of the Yuan as a legitimate dynasty. Curiously, he also decries the Mongols for bringing the "rank odor of sheep's flesh" into China; i.e. corrupting China with their barbaric customs. The comparison, tension, and conflicting nature of early Ming rhetoric and early Ming policy is actually super fascinating and is part of my dissertation.

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Sources:
Rossabi, Khubilai Khan: His Life and Times
Rossabi, "The Reign of Khubilai Khan," in The Cambridge History of Chinese History, Volume 6
Endicott-West, Mongolian Rule in China
Dardess, Conquerors and Confucians
Mote, Imperial China