The Mongols had a few things going for them too. Their armies were awesome to behold and they moved as a nation. That means that almost all men were warriors and that gave them the advantage of numbers. By that I mean a larger army. Most armies are based on 1% or less of a countries available manpower, I don't mean to denigrate women here but back then it was manpower. When a country really mobilized it might get 10%. The Mongols were at 50% or more. Add top that the mobility of their armies as compared to China and you see that they had advantages. They also had a few great leaders both as tacticians and strategists. Add those up, and down goes China. Of course, it took quite a bit of time and multiple invasions/campaigns to complete the conquest.
More than anything else, I think the Mongols were terrifying. They didn't fight "Fairly" by modern standards. If they came across a settlement with tall walls, they would resort to whatever methods necessary. When I was in school, I remember a teacher telling me about how they would hurl the bodies of those that died of deadly diseases over the walls, exposing the city to the sickness. Eventually, it would spread so much that the civilians would surrender and the city would be lost. When you heard that the Mongols were headed towards you, it was terrible news.