Some brief background first: Cyrus was a Chalcedonian bishop appointed as the Patriarch of Alexandria and effectively governor of Egypt by the Emperor Heraclius in 630. Egypt was however largely miaphysite rather than Chalcedonian, and had its own ecclesiastical hierarchy, which survives to the present day as the Coptic Church. The split between Chalcedonians and miaphysites originated from the Council of Chalcedon in 451, and since then Roman emperors tended to favour Chalcedonianism (Anastasius and Justinian I being two major exceptions), with the result that there were several waves of persecution against miaphysites over the next two centuries. There were also of course more local battles between miaphysites and Chalcedonians (often initiated by monks for some reason), and inevitably internal struggles between different strands of miaphysitism. There was also a long history of compromises and detente between the two sides, so keep this in mind as well whenever you read about Copts being massacred by the Romans.
So why did Cyrus decide to 'massacre' the Coptic Christians? To answer this, we have to talk about Heraclius. Based on the sources, it appears that Heraclius had tried throughout his reign to reconcile the divided Christian communities within his empire. One of his earlier appointees to the Chalcedonian Patriarchate of Alexandria, John the Almsgiver, actually played a role in reconciling the Egyptian and Syrian branches of miaphysitism with each other in 617, even though Heraclius and John were both Chalcedonians. After Heraclius' miraculous victory over the Persians in 628 he embarked on an even more ambitious project of reunification. One of his policies was the forced conversion of Jews to Christianity, another was the promotion of monoenergism and later monotheletism, two doctrines that were seen by older historians as a way to bridge the divide between Chalcedonianism and miaphysitism, though more recently it is seen as a natural evolution of Chalcedonian beliefs. Either way, Heraclius sought to impose this on his subjects, a policy that received support from the Patriarch of Constantinople, several major miaphysite leaders and even Pope Honorius I.
Cyrus was one of the first bishops to agree with this formulation and it is generally assumed that he was awarded by being sent to Alexandria. From Coptic sources, he then allegedly began a campaign of terror against Egyptian miaphysites to convert them to monotheletism, in the process forcing the Coptic Patriarch Benjamin to flee into exile and martyring his brother (after lengthy torture of course), to name just two examples. However, this represents only the views of later miaphysites and in reality the situation was probably less dramatic (though much more interesting). In 637 Cyrus actually managed to get the local miaphysite hierarchy to sign up to his beliefs in the Union of Alexandria. The Coptic sources say that they were forcibly converted, but a letter from Cyrus to Constantinople reported that most of the bureaucracy/tax-paying population was now in communion with the wider church, even many monastic communities, who were often the most vocal opponents of Chalcedon.
I'm sure Cyrus did some pretty terrible things, as I doubt Coptic writers made them up out of thin air, but the persecution was most likely targeted at intransigent leaders (Benjamin's family for example) and did not lead to any major massacres - military resources were definitely better spent against the Arabs rather than the population (the majority of whom were miaphysites, so it would have been quite difficult for Cyrus to persecute all the Copts). Later records of Cyrus' involvement in negotiations with the Arabs also paint him as a very human figure trying to save Alexandria from a bloody siege by negotiating with 'Amr after several major defeats. I think therefore it is quite likely that the persecution of the Copts was exaggerated by later sources (which are almost exclusively miaphysite in character) once Egypt was firmly miaphysite again and Cyrus had been disgraced after the fall of Egypt to the Arabs. After all, the same was true for the 'proper' Chalcedonians - pretty much all major Chalcedonian leaders signed up to monotheletism in the 630s, but sources written after 680 (when an ecumenical council condemned it) blamed only a few ringleaders and damned the doctrine as a vile heresy, when from earlier sources it is clear that most people were perfectly okay with it.
Feel free to ask me to expand on any of this, not often I get questions about heresy in the seventh century, one of my favourite topics :)
Recommended reading:
John of Nikiu, Chronicle - a very important source from a miaphysite writer who lived only a few decades after the Arab Conquest, naturally quite biased against Cyrus.
Severus, History of the Coptic Patriarchs of Alexandria (I linked directly to the section on Patriarch Benjamin) - a much later miaphysite source, but it is believed to incorporate earlier material as well, so this provides another anti-Cyrus perspective.
Booth, P., Crisis of Empire: Doctrine and Dissent at the End of Late Antiquity (Berkeley, 2013) - the best book by far on the church in the seventh century and provides the most up-to-date information on all the doctrinal conflicts during this period.
Allen, P., Sophronius of Jerusalem and Seventh-Century Heresy: The Synodical Letter and Other Documents (Oxford, 2009) - not strictly about Egypt, but Allen's translation of important documents about the monothelete controversy is very useful for getting a sense of the context of the crisis.