This is quite vague, so I'll give you a little bit of the economic, military, and imperial reaction. First off I'll give you a brief overview of the situation in Byzantine Egypt before the Islamic conquest. In short the peoples of Egypt did not like being under Orthodox Christian rule this goes back to the Monophysite/Monoenergism/Monothelitism schism. The first originated under Theodosius, and the latter two under Heraclius, but long story short they really didn't like each other, and the Egyptians especially didn't like Emperor Heraclius for both religious and moral reasons. In addition to chafing under imperial rule Egypt was also suffering from plague and famine at the time of Arab conquest. As such many local Egyptian leaders were willing to work with the Arab conquerors and made the conquest of Egypt that much simpler. Around the time the majority of Egypt, sans Alexandria, fell the imperial army had declined from its height of 150,000 men in 602 to around 109,000 in 641 and the Emperor had passed away from a combination of age and dropsy. He was succeeded by his son Constantine III who promptly died and was succeeded by Constans II who would preside over the Byzantine response to the Arab conquests. Before there could be a response however, there was a bit of a succession crisis which prevented any real Byzantine retaliation to the conquest of Egypt, and Egypt fully fell in the autumn of 642. In 645 the new emperor sent a rather massive expedition to recover Egypt, but was repulsed, this would be the last time the Byzantines really had a chance at recovering Egypt. At this point the Arabs began to press the remaining provinces of the Empire with renewed force, and the Exarch of Africa declared himself emperor. At the same time the various popes, patriarchs, and bishops of the many flavors of Christianity in the Empire began to once again bicker over the nature of the Trinity. despite the imperial edict that forbade just such discussion.
I will, however, skip over all that stuff and get the the real economic and military reform of the Theme System. In an attempt to hold on to what land remained to the empire and maintain an army that was comparable to the army maintained by his grandfather Constans II implemented the Theme system (This is a matter of slight debate as to whether it was Constans II or his successor who implemented the Theme system, my sources tend to argue in favor of Constans II). This system did quite a lot, it halved the soldier's pay in return for hereditary land grants, the empire no longer armed and armored its soldiers (they were expected to do that themselves), and this had the unintended side effect of dodging the idea of feudalism almost entirely and introducing the concept of private property centuries before the idea would be rediscovered in industrialized England. In addition to a militarized provincial system bands of fortification were built across Anatolia to provide defenses from Islamic raiders and border lords known as Akrites were put into place along the Taurus and Antitaurus mountain ranges. All of this did not happen immediately, and some of this, e.g. the Akrites, did not happen under the reign of Constans II.
From a true economic standpoint all I can really find is the end of the free bread dole in Constantinople, and the decline of Mediterranean trade as piracy dramatically increased.
Dates and Numbers from Warren Treadgold's History of the Byzantine State and Society
Herrin, J. (2009). Byzantium: the surprising life of a medieval empire. Princeton University Press.
Treadgold, W. T. (1997). A history of the Byzantine state and society. Stanford University Press.