The Ethiopian church (specifically the Ethiopian Orthodox Tewahedo Church) is actually one of the oldest groups to split off from the established church all the way back in the 5th century. They're actually completely distinct from the "Eastern Orthodox" Christianity of, for example, the Greek and Russian orthodox churches, but are their own thing called "Oriental Orthodox" or Miaphysite. In fact the split between the oriental orthodox and the broader church at the time actually predates the East-West schism between orthodoxy and catholicism by about 500 years.

The initial split was over a rather obscure theological issue over whether Jesus was of two natures, one human and one divine (the position held by the Catholic church) or one nature that is both human and divine (the position held by the Oriental Orthodox). This might sound incredibly nit picky and esoteric now but it was taken incredibly seriously by early Christians because the simple issue of who or what exactly Jesus was was not totally settled at the time. The concept of the trinity as we know it today was one of many competing ideas like Nestorianism (Jesus Christ and the Son of God are technically distinct entities), Adoptionism (Jesus was just a man until God decided to grant him the status of a God), and even Docetism (Jesus was purely a divine spirit and his physical body was an illusion).

Although the issue is considered purely semantic by theologians today, the issue of defining the nature of Jesus in exact detail was something that people at the time were really not willing to compromise on. When the issue was raised at the council of Chalcedon in 451, tensions got so high that a permanent split was caused. There's also some evidence that this also had a lot to do with what were basically political disputes over the authority of various bishops and what the borders of their domain should be. There were limited attempts to reconcile over the next hundred years or so but they were unable to come to an agreement and at that point the Oriental Orthodox had already started to establish its own structures that were used to operating independently.

In any case the Oriental Orthodox churches were initially a pretty huge deal. They included not just the Ethiopian Church, but also most Christians in Egypt, Syria, Armenia, and elsewhere. This is before the spread of Islam, so Egypt and large parts of the Middle East were majority Christian at the time. Oriental Orthodoxy lost a lot of its influence when it was displaced by Islam during the Arab Conquests of the 7th century, and even more so by the Ottoman Empire later on. Despite this the majority of people in Ethiopia, Eritrea, and Armenia all follow Oriental Orthodox denominations today, and there's pretty sizable minority communities in Egypt, all throughout the Levant, and even in India. Today there are 6 recognized Oriental Orthodox groups spread around Africa, Asia, and the Caucuses, and while they've all evolved their own unique practices due to limited contact over the centuries, they are all considered to be in communion with each other.