It's complicated! I would like to point you to an old answer of mine, if you've already read that/have any follow up questions please let me know, I'd be happy to expand.

I can answer this for China! I'm mobile so if you'd like sources I can provide them at a later time.

In China, several dynasties were really obsessed with purity of the royal lineage. For practical reasons this led to the famous harems of the Forbidden Palace in later dynasties such as the Ming and Qing. The concubines of the emperor were served in their quarters by outer guards (male and not eunuchs, and not really in the rooms from day to day), women servants, and eunuchs.

Eunuchs were good for sending messages, household records, and so on. Some held political official positions but this practice was largely banned by the Ming. I don't know much about their use before this, but I can speak to the Qing era more clearly since it's my primary field of reading.

By the Ming dynasty most eunuchs were volunteers. I suppose this was not a bad way to make a living and in fact many eunuchs enjoyed households and monetary wealth outside the palace. In the Qing era, all the eunuchs would have been Han as the Manchu elite were forbidden to volunteer for this. In this era I think the Manchu/Han difference cannot be overlooked in a discussion of power dynamics.

In Chinese culture, castration meant you weren't a whole person. Some believed that if you were not buried with your genitals, your afterlife would be altered or incomplete.

In many ways it was easy for eunuchs to become confidants of concubines and emperors since they served intimately and would have been assigned to their masters or mistresses individually. There are a few in the Qing dynasty who "overreached" their job descriptions, so to speak, and gained significant power as advisors to the rulers.

Many of these powerful eunuchs reached those positions after having served the emperor (or empress, in one famous case) for decades, sometimes since both were basically children or adolescents by today's standards. In a world that is very small socially where people had little contact with friendly faces, and many political and familial factions were at work, it makes sense that eunuchs became trusted servants. Because they ran more errands outside individual quarters and the network was large (entire departments managing the harem and lineage records were run by eunuchs) with its own factions and affiliations, eunuchs could get pretty involved in major decisions and moves in the lineage and the choice of the next emperor. And plenty of uprisings and civil wars were fought between siblings for the throne, and eunuchs would've been right in the thick of it before any fighting even started.

But for the most part, men who were castrated for this purpose became second class citizens who had to earn their genitals back to be buried whole. Relatively few made it to notoriety.

Edit:

Sources: 《明史·宦官傳》《清实录》, and this real fun one about eunuchs and female binding in Chinese civilization: http://www.huanqiu.com/special1/book/beiyangedewenming/ (online text of published book). The title is kind of gross (in Chinese).

Eunuchs weren't equally popular around the world. They were much more common in the East, then in Western Europe. I can give some Byzantine examples of why they sometimes achieved high positions.

Eunuchs were a commodity, created to exist for several purposes. This included several areas, including as attendants for women were men would not be appropriate in the culture. They were also often trained as bureaucrats. Why? Several reasons.

The Empress Irene had many high level eunuchs, including her the chief advisor and several as the heads of her major armies. The reason she had eunuchs was so a man would not try to take over her position. A position she gained after she used her Regency to suppress her son, who briefly became Emperor in his own right, but was later controlled by Irene and her eunuchs and then blinded and killed by them.

This blinding of her son shows some of the reasons why eunuchs were important in Byzantine culture. Some physical mutilations culturally left you unqualified to be Emperor. This is why Emperors sometimes had rivals' noses slit, so that the physical deformity would prevent them from seeking the crown (didn't always work). Being a Eunuch meant you didn't have much of a chance of being accepted as emperor by the people. Thus, you were a safer ally to have at the top. Irene did not want some man trying to usurp her power, thus she employed Eunuchs.

Justinian employed a Eunuch named Narses to lead a grand army during his campaigns in Italy. Great generals sometimes had the habit of using their victories to proclaim themselves emperor. This is less likely (and from what I know, never happened) with a eunuch. His use of Narses very likely was a response to his other general Belisarius, who was very capable and made Justinian fear for his throne.