The question is less mysterious if we rephrase it as "How did tribes of sheep herders and breeders of an essential military resource, horses, bring down a nation of farmers over a time of 68 years?"

Q: How were the Mongols able to conquer China?

A: They did it by building up large armies with plentiful siege equipment, including artillery such as catapult-thrown gunpowder bombs, and the skill to use it, and a powerful navy. All of this depended on using Chinese manpower and Chinese industry.

Q: How were the Mongols able to make use of Chinese manpower and industry?

A: By conquering part of China.

Q: How were the Mongols able to begin conquering China?

A: They joined in an ongoing war for the control of China. When the Mongols joined the war, the major powers were the Song, controlling southern China, the Jin (Jurchen), controlling Manchuria and much of northern China, and Xi Xia (Tanguts) to the west of the Jin.

These wars had been in progress since 976, starting with fighting between the Song and the Liao (Khitans). The Jin overthrew the Liao, and fought the Song from 1125 to their final defeat by the Mongols in 1234.

Before beginning their war with the Jin, the Mongols attacked Xi Xia (1209-1210), and Xi Xia submitted and became a Mongol vassal/ally, contributing troops to the fighting against the Jin. In 1211, the Mongols began their war against the Jin. The Jin faced not only the Mongols, but also Xi Xia, the Song, and rebellions in both Manchuria and northern China. By 1215, the Mongols had conquered the northern Jin territories (the Jin moved their capital south to Kaifeng, and the Mongols captured Beijing in 1215). With the resources of northern northern China and Manchuria (and Mongolia, and Xi Xia), the Mongols were able to continue their war against the Jin.

The final conquest of Jin was delayed by the Mongol invasion of Kwarezmia, but resumed fully in about 1230. The fighting between the Mongols and the Jin hadn't stopped between 1215 and 1230, but had been sufficiently low-level for the Jin to engage in a major (unsuccessful) offensive war against the Song. The Song gave important aid during the last Mongol campaigns against the Jin - this included fighting against the Jin, but perhaps most important was logistic support for Mongol armies in southern Jin which had been devastated by years of warfare and the Mongols were not capable of sustaining their armies their without food supplies from the Song.

With the defeat of the Jin, the Mongol had the resources of all of northern China and Manchuria to use for their war against the Song. A powerful navy was essential for control of the Yangtze river, which was essential for success in the fighting to break the main Song defensive line along the Yangtze. The key battle for breaking the Yangtze defences was the five year siege of Xiangyang.

This first part of the war, breaking through the outer Song defences, took from 1234 to 1273 (the Siege of Xiangyang was 1267-1273). After this, it took a mere 6 more years of warfare, including naval battles for the final defeat of the Song.

One point about manpower: While Chinese states often had very large armed forces (e.g., 1-2 million), these soldiers were dispersed over a large area. First, this was to protect a large area from outside threats and internal rebellion. Second, logistics limited how much of that total army could be concentrated in one place. If we add to this Mongol superiority in operational mobility, the Mongols were able to defeat the Jin in battles because they could fight with local numerical and/or qualitative superiority, even if the total Jin forces outnumbered them. It isn't necessary to defeat the entire Chinese army, especially not all at once.

For an excellent coverage of the Mongol-Jin war, see Carl Fredrik Sverdrup, The Mongol Conquests: The Military Operations of Genghis Khan and Sübe'etei, Helion (2017). For the Mongol-Song war, see James Waterson, Defending Heaven: China's Mongol Wars, 1209-1370, Frontline Books (2013).