Some form of slavery persisted in China through much of imperial history, but it was so small a part of society, with such minute numbers that it is difficult to prop it up and say: "look! Slavery is there too!" This element was the eunuchs, but by the Qing, the eunuchs were less and less captives of foreign lands, and more and more either voluntary or essentially sold into the eunuch system by middling families looking for an imperial connection.
So, with this tiny asterisk to the history of the lack of Chinese slavery behind us, let's look at why slavery died out in China:
It persisted in the Han dynasty as an outlet for the overflow of enemy combatants in war, and as a place for society to put some of its discontents, those who committed crimes or owed a lot of money. But the numbers weren't very high, and the possibility of remission was open.
Why did even this low level slavery end? It is hard to say. I haven't read any good scholarship on this question, so I can't say much more other the this:
Slavery was never common, never household oriented, and never became involved in any major industries, and I think it was simply impractical, creating an undesirable class that didn't really jive with Confucian society. there was simply no good Confucian rationale for them (although there was, by contrast one for eunuch-slaves in the imperial house).
However, this slavery in China (and its lack) was not much different than slavery in many other places before the massive scale of the American slave system buried all the slaveries of the past.