I think the success is an impression. Because of China's myriads of historical peasant rebellions, only two can be called successful - the Red Turbans who overthrew the Mongol Yuan Dynasty and established the Ming Dynasty, and the communist revolution which established the PRC (and even that one, you can argue either way because the PRC did not start as a peasant based organisation, it just became that way when the shift in communist support happened after the White Terror).

The other famous peasant based rebellions - the Yellow Turbans, the Kingdom of Heavenly Peace, the Chen Sheng rebellion at the end of the Qin Dynasty...did not eventuate in stable dynasties.

As for why they're so present, I don't know of established historical theories so I'll let others take a shot at it first. I'm trying to formulate an argument about how the difference between mandate of heaven vs. divine rights of kings may affect this.

edit: okay I've come back and am forced to conclude that the mandate of heaven system and relative meritocracy of feudal China vs. medieval Europe must have a role. Particularly, the mandate of heaven is a system which means that the Emperor rules by the will of heaven but this will is not eternal and can be withdrawn. This is usually tied up in the Confucianist system of morality where the personal virtue of the Emperor/his predecessors affected the mandate of Heaven.

So basically things like natural disasters, hardships etc. are a sign that the mandate of heaven for the dynasty has been withdrawn, and uprisings can in the eyes of rebels be justified that way. The Records of the Grand Historian by Sima Qian records that the anti-Qin rebels found a fish with a message in its gut saying that the Qin Dynasty must be overthrown. Records of the Three Kingdoms by Chen Shou also records the Yellow Turban rebels claiming portents etc. for the end of the Han Dynasty, and these claims/rebellions generally coincide in times of hardship/misrule. So this fallibility of the Imperial clan is one factor which leads to rebellions being seen as okay.

The second factor then is that starting from the Eastern Han Dynasty there really wasn't that much in China in the way of nobility - ie. princes, dukes, barons and what have you. There were wealthy land owning families, scholar families (the gentry), merchant families etc. but really you couldn't call them feudal nobility like you could with the European aristocracy. So with this kind of semi-meritocratic society there is no need as there is in Europe to find alternative monarchs based on bloodlines and claims etc. Of course it generally helped if you could claim you were from a better family. eg. in the Three Kingdoms era the three competing dynasties had their founders descend respectively - from the adopted son of a eunuch (not something very respectable), basically a commoner (who had ties to the royal family like a century ago) and from a merchant (again, not very respectable).

Impression is one of them. The unification of China goes all the way back to the time of the Roman Republic. That's a lot of history compared to the relatively short history of modern European nation-states, which makes for a lot more opportunities for successful peasant rebellions.

Also, in the post Roman world, Europe was far more fragmented compared to China. Whereas China maintained a relatively unified culture, government, language, etc., Europe was splintered into many different feudal states with varying cultures, language, and government.

Keep in mind that China has historically encompassed an area the size of Europe and a population to match as well. A peasant rebellion in 800AD in say, Normandy, may not be much of an concern beyond the Duke of Normandy and may have little influence on a county in Burgundy which at the time would have shared a different kingdom entirely. However, in that same time, a peasant rebellion in Hubei could ignite one in Hunan and spread across China who were all under the rule of the same Emperor.

It could have been related to the "equal field system" that was set in place as a tax system. Land was distributed equally to cultivators (males who were able to farm), who paid taxes in labor and grain. However, because the scholar/elite class were exempted from paying taxes, yet were able to gain greater amounts of land through "grants" from the emperor, they levied this heavy tax weight back onto the peasant class. The system was thus unequal, times get hard, and the peasant class would rebel.

Anyway, this is the theory I remember from my Chinese history class. Might not be 100% correct but I am sure it is along these lines. I double checked this theory from the source below.

Source: Craig, Albert M.. "The T'ang Dynasty." The heritage of Chinese civilization. Upper Saddle River, N.J.: Prentice Hall, 2001. 58-59. Print.

Supplementary question here, how are peasant rebellions represented in Chinese literature and popular media historically speaking?

It is mostly an impression, although they were never successful, and with the success of the Red Turbans in China no Chinese peasant revolt was successful either, they were a relatively common occurrence throughout medieval Europe. The Jacquerie in northern France during the Hundred Year's War stands out as one notable example, but was definitely not the only one. Also the Great Rising of 1381, in which Richard II of England was nearly overthrown was another mass peasant uprising that nearly succeeded. Alastair Dunn has a great book, The Great Rising of 1381: the Peasants' Revolt and England's Failed Revolution which goes into a lot of interesting depth about the revolt. Finally throughout the Reformation large peasant uprisings, culminating in the Anabaptist uprising at Munster in 1534 occurred. Throughout periods of long warfare, high taxation and military defeat all through European history peasants rose up to try to break the status quo