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A brief, oversimplified explanation is that they are what is left of those Christians who rejected the Third Ecumenical Council. They only accept the first two. Their reasons for rejecting the Third Council is that it condemned one of their Church Fathers, Nestorius, as well as his Christology. Nestorius' Christology, also called Theodorean Christology (after Nestorius' teacher Theodore of Mopsuestia,) treats Christ's divinity and humanity as if they function separately. Again, this is a really oversimplified explanation. The Assyrian Church of the East still venerates Nestorius and Theodore as saints, and upholds their Christology. Back during the days of the Persian empire, the Church of the East was referred to as the Persian Church, as the Assyrians existed mostly behind the Persian border.
When they separated from the Church is a complicated question. The easiest answer is to say they separated after the Third Council, but there were instances of reconciliation after that. After the Third Council, there was a formulary of reunion between St. Cyril (who presided at the Third Council) and John of Antioch, who represented those who did not want to sign onto the Third Council. However, that was a very tenuous agreement which was subject to different interpretations and it soon fell apart. It can be said that the predecessors of the Oriental Orthodox finally and completely parted ways with the Theodoreans at a council held in Ephesus in 449.
The history between the Assyrians and the Chalcedonians (EO's and Catholics) is a little more complicated. The Assyrians interpreted the Council of Chalcedon (held in 451) in a Theodorean manner and accepted it, so it can be said that the Assyrians were in communion with the Chalcedonian Church for at least a century after Chalcedon. However, in 553 the Chalcedonians held their Fifth Ecumenical Council, at which Theodorean Christology was unambiguously rejected. So at that point the Chalcedonians and Assyrians would have been out of communion. However, in the late 500's, the emperor maurice reunited his Church with the Assyrians based upon a statement of faith submitted by the Assyrians (I don't think they accepted the Third Council.) So for a while they were back in communion. I'm not sure when that agreement fell apart, but I don't think it would have survived very long after the fall of the Persian Empire to the Muslims.
I'm not a historian and, like I said, this all oversimplifies things. If, however, I made any glaring mistakes, I am sure someone will be kind enough to correct me.
