I’m going to frame my answer under the assumption that, because you asked specifically about horse archers, what you mean by “successful” pertains directly to military success. The answer goes beyond that but I’ll start there.

The first thing to know, is that after 1206, Chinggis and his commanders began including outside military units into their armies as well, initially northern Chinese, Tanguts, and Uighurs. This happened because they were beginning to expand beyond the boundaries of Chinggis’s initial steppe conquests. However, the same approach he took for enveloping conquered nomadic tribes into his empire he also applied to conquered sedentary populations (initially, under Kublai, for example, this would change), promoting the meritorious and loyal among them and redistributing property to those loyal meritorious troops. So this should answer your initial question about horse archers. While a key piece of the military, other units like heavy lancers and siege engineers were also critical to the Mongol destruction of the Khwarezmian empire in the 12-teens and beyond. A good example of this merit and loyalty matrix of promotion can be seen in the figure of Gao Kan, a Chinese general present at the fall of Baghdad. Gao (surnames come first in Chinese families) also was the governor-general of Baghdad for a time, indicating that during the life of Chinggis and his immediate successors, merit and loyalty were key to one’s elevation. Gao’s father and grandfather had also served Chinggis directly, an indication that these positions could be perpetuated along family lines once established.

That’s enough about horse archers for now, because there were also social aspects to this. The Mongols arose from a situation in the steppe of different nomadic tribes in competition for supremacy over one another. Sometimes one or another was successful at coalescing enough power that they could “unify” the steppe for a brief period in a confederation. The xiongnu are a good example from the Han period. Chinggis’s great grandfather had been one of these leaders - a khagan, or khan of khans - but the problem was that these confederations usually dissolved with the death of the charismatic powerful khagan who amassed them, returning the steppe to the cycle of competition. With Chinggis’s conquest of what would become Mongolia, he made efforts to destroy this phenomenon (whether aware of it or not) by investing power of the khagan only in his progeny, rather than another tribe or even other members of his own Borjigin clan. In following the model of a nomadic confederacy and then taking steps to assure continued power for his sons and daughters and grandchildren, chinggis reshaped that same pattern to develop inertia in his family’s power, thus in some ways deviating from the pattern of the nomadic confederacy cycle of unity and competition.

Returning to military, we have evidence that elite Mongol women displayed a propensity for very competent administration as regents while their husbands, fathers, and sons were away on campaign. Much more-so, because Mongol men had a virtually 100% participation rate in the military, women often governed affairs of the state and household. In elite circles this meant literally governing their own fiefdoms given to them by their male relatives and where they had power in their own right. While Mongol perspectives on women and violence toward women were far from enlightened, households also displayed a high degree of female autonomy. While also a display of an imbalanced division of labor, Mongol women took charge of moving camps and for the logistics therein, which gives a hint to the high degree of autonomy they had outside of the male patriarch’s ger. Again, do not mistake this relationship as being some utopian and egalitarian system (Crash Course World History helps perpetuate that myth). Non-elite women definitely got the short end of the stick in Mongol society as with others. However, in terms of holding power at home while the majority of the fighting force was away on campaign, Mongol women were instrumental and capable administrators.

I have to go to work but if there are other questions, I haven’t really talked about the diffuse style of Mongol governance over their conquered lands. (long story short in a lot of places they left local power holders in place who agreed to pay tribute to them and so long as that relationship remained stable, Mongol elites were happy. See: Alexander Nevsky and Hethum I of cilician Armenia for examples.) One could also discuss Mongol tactics as examples of this success and the initial European campaigns are evidence of how powerful the caracole and Fabian tactics they mastered could be. But, by the late 13th century, Europeans had learned how to deal with them and were much more prepared the third time the Mongols came back through Ukraine. There are many more routes to examine with this (some people even take the environmental determinism route of pointing toward the higher grass yields of the Mongol pluvial that occurred at the start of Chinggis’s life)—ultimately there isn’t any one thing. Other responders can do a great job of adding onto this if they wish.

Some accessible sources:

De Rachewiltz’s translation of The Secret History of the Mongols is available online and gives some good ethnographic info on steppe life in Chinggis’s early days.

George Lane. Daily Life in the Mongol Empire,2006.

David Morgan. The Mongols, 2nd Ed. (I can’t remember the publishing year for the 2nd Ed., sorry)

I believe Ata Malik Juvayni’s History of the World-Conqueror is also available many places online too. A good look at a 13th century history of the Mongols by a Persian administrator in the empire.