Why did it reach further east than Abrahamic religions? Well, it didn't. "Nestorians" got as far as China (twice, in the Tang dynasty until they were purged in the Great Buddhist Purge, then again under the Mongol Yuan Dynasty), which is the same distance east as Muslims got.

If you mean why did it get further than Catholics, it's important to remember where the division started. In the 3rd and 4th century, Christianity had spread west into Europe and east into Persia. What dramatically changed is that the Roman Empire adopted Christianity, first simply as the personal religion of the Emperor in the early 4th century, later as the state religion. Within a century, the majority of Romans were (at least in public) Christians. As the new centre of Christianity, Rome began to hold councils to decide, once and for all, exactly what Christians believed. At one of these councils, Nestorius, Patriarch of Constantinople, was condemned for heresy and stripped of his position. However, many of the non-Roman Churches in Mesapotamia and Assyria supported him (some because they agreed with his views, others didn't but still supported him). So his followers re-located to these Churches, which became known by westerners as "The Nestorian Church" but which called themselves things like "The Church of the East" or "The Syriac-rite Church" (specifically, Ēdtāʾ d-Maḏenḥā). Some of the Christians agreed with Nestorius, some didn't, so it can't be said the Church was really "Nestorian" in doctrine, just that it was a tolerated opinion.

Under the Sasanians (Persians) the Christians became fairly organised as a minority religion, and the Church of the East was recognised as a distinct Dhimmi religion after the Muslim conquest. In the 6th and 7th centuries, they sent out lots of missionaries east and north, to Central Asia, India and China. These missionaries to Central Asia are the ones whose beliefs eventually spread to certain Mongols. However, what the Nestorian Church lacked was the support of a strong state. The Roman Church was the state Church of a mighty Empire; the Nestorian Church was a minority religion under first the Zorastrian Persians and then the Muslims. The Persians had given them some support (they tended to use the Nestorian Church as a proxy in their Arab buffer states, while the Byzantines used Monophysite Arabs) but after the Islamic Conquest, they were recognised but not supported.

So, while the Nestorians did spread Christianity very far, into lands that had never heard of Christianity, they never managed to convert huge numbers of people.

Also, they faced various, effective purges. They were driven out of China twice, once in the Great Anti-Buddhist Purge of the Tang Dynasty, once by the Ming. When the Mongol Empire fractured, a number of its leaders converted to Islam, and one, Timur, killed most of the Christians in Persia (he also killed a ton of Muslims). At that time, most people in Central Asia converted to Islam.

So, essentially, the Nestorian Church got furthest east because it was the eastern Church. The Catholic Church got the furthest west because it was the western Church. To explain that more clearly: Western Christians evolved into the Orthodox/Catholic Churches, Eastern Christians evolved into the Nestorian Church. However, the Nestorians lacked the state support and mass numbers to survive in the long run.

These days, they are a small and fractured community, with small numbers in India, Iran, and the Middle East. They are also split between two organisations: the Assyrian Church of the East, and the Chaldean Catholic Church (which has joined the Catholic Church, but still keeps its traditional Syriac rite and traditional services, rather than adopting the Latin rite).

There is an excellent book by Lev Gumilyov - Searching for an Imaginary Kingdom : The Legend of the Kingdom of Prester John (1970). Although it is primarily about Mongols, it also puts emphasis on analysis of religious situation leading to events of 13th century, including Nestorianism.

Based on this (and other sources), I'd like to correct few facts.

a number of Mongols and other Steppe tribespeople were Nestorians

First of all, Mongols were certainly not entirely Nestorians. Mongol khans were always dependent on their army and since they had practice of incorporating surviving enemy units into their army, they could not openly practice any religion, really, and certainly not Nestorianism, which was extremely aggressive towards other religions (See the historical accounts of monks Plano Carpini (1246) and William Rubruck (1253), who were sent from Europe by Pope to negotiate with Khan and observe the situation there).

Mongols' original religion was Tengriism, which is sort of dualistic shamanistic religion with Heavens and Earth as their main deities. Genghis khan himself practiced this religion, as it is said he "prayed" to them in the mornings and before battle (Secret Chronicle of Mongols, ancient source).. The Great khans were mostly indifferent to religious views of their subjects, which was essential for their various soldiers and also keeping of order in vast territories. (There is also distinction between Great Khan and his subordinate princes, such as Hulegu (who was inclining to Buddhism), Kublai khan, who was close to Christianity but later abandoned it when he conquered Chinese throne and become first foreign emperor of whole China, Sartaq and later his son Uzbeq, who later converted to Islam and thus making Golden Horde part of the Empire muslim. These various religious affilations later cause civil wars among Mongol Uluses, for example Golden Horde and Ilkhanate)

How did this reach farther east than other Abrahamic religions?

Even after being deemed as heresy by the West, they survived in the Middle east (mostly under Persians) and some of them fled to east from muslim invasions, where they were met with religious tolerance. I would argue that is not farther east than other Abrahamic religions, as there are reported skirmished between Tibetians and Muslims, and the Muslims did also reach India and China...

it appears that Nestorians and Catholic were in rather good terms

Nestorians and Catholics certainly weren't in good relations. Council of Ephesus (431) condemned Nestorianism as a heresy and it later proved crucial when Mongols arrived to Middle east in early 1250's.. After their Great khan Monghke died in 1259, most of mongol armies returned to their homeland to vote for a new Great khan, and Crusaders, then situated in what remained of Christian states in Levant, decided to aid the muslim Mamelukes against Mongols. It has to be said that Crusaders and the West indeed had knowledge of Nestorians in mongol ranks and some (Armenian King Hethum and Antioch's prince Bohemond) even decided to side with Mongols for some time, but at the time heretics were seen as even worse than Muslims... Also the in Europe's fresh memory was the Mongol invasion of Hungary and Poland (1240-1242)..