First: a great sum of the Christians in Egypt at the time of the Arab Conquest were the forerunners of the contemporary Coptic Church. This identity goes back to a split in which the Patriarch of Alexandria and the Church there rejected the Christological formulation established at the Council of Chalcedon.

This led to a period of struggle between those in Egypt who rejected this Council, and the Church figures that were put into place by Byzantium. This continued until, as said before, the Arab Conquest of Egypt. Those who rejected the Chalcedonian Creed had no reason to support Byzantium, and were given three options: fight, convert to Islam, or pay a tax (jizya). The third was more acceptable choice, especially among the non-Chalcedonian clergy, for it would allow them to regain the Patriarchate of Alexandria.

A good three volume set that covers the Coptic Church:

• Davis, Stephen J. The Early Coptic Papacy: The Egyptian Church and Its Leadership in Late Antiquity. American University in Cairo Press, 2005.

• Swanson, Mark S. The Coptic Papacy in Islamic Egypt. American University in Cairo Press, 2010.

• Magdi Guirguis and Nelly van Doorn-Harder. The Emergence of the Modern Coptic Papacy. American University in Cairo Press, 2011.

The information I posted namely comes from the first volume in this set.