Short answers, pulled from the story.
Babur was a direct descendant of Genghis Khan through Genghis Khan's son Chagatai Khan. This Mongol lineage, combined with his Timurid ancestry, meant Babur was descended from two powerful dynasties simultaneously.
The Mughal rulers shared genealogical relations with Mongol royals at a time when such distinctions had become uncommon. Their direct descent from Genghis Khan through Chagatai Khan, and their Timurid heritage, placed them within the tradition of Mongol successor states.
In 1370, Timur married Saray Mulk-khanum, the daughter of Qazan, the last Chaghatai Khan of Mawarannahr, after destroying the army of Husayn of Balkh. This marriage made Timur an in-law of a direct descendant of Genghis Khan, and he adopted the title Temur Gurgan, meaning son-in-law of the Great Khan.
The Mughal Empire descended from the Timurid dynasty, through Timur (Tamerlane), and from the Mongol royal line, through Genghis Khan's son Chagatai Khan. Both lines converged in Babur, the empire's founder.
The Tarikh-i-Rashidi is a history of the Moghuls of Central Asia, edited and translated by Elias and Denison Ross, first published in 1898 and reprinted in 1972. It is one of the key historical sources for tracing the genealogical connections between the Mughal rulers and the Mongol Khans.
Through their Timurid ancestry, the Mughals were connected to royal families across the Mediterranean, the Middle East, and the Far East. Related dynastic genealogies also link them to the Safavid dynasty of Persia.