It didn't; Egypt was never Shi'a. They were ruled over by a Shi'a dynasty, the Fatimids between 969 and 1171, but the Fatimids never imposed their religious views on the population and, consequently, the majority of the population remained Sunni.

How and why this happened ... or, rather, how and why this didn't happen ... is still the subject of considerable discussion among historians. It's worth pointing out that the whole concept of Sunni and Shi'a were still relatively undefined at this point (the Fatimids were Ismaili, which at any rate is a minority branch within Shi'ism; their ideological descendants are the followers of the Aga Khan today).

One of the prevailing theories is that the Fatimids were trying to demonstrate a concept of universal rule -- that is, that they were ideal Muslim rulers who could rule over all Muslims regardless of their particular ideological bent, and this way unite the Muslim population in a way that hadn't been done almost since the days of the Prophet Muhammad himself. Their ideological rivals, the Abbasids, were less tolerant of dissenting beliefs, so the Fatimids made supporting the underdogs part of their mission: "everyone has a home here, with us."

The third chapter of Christine D. Baker's Medieval Islamic Sectarianism is (despite the title) a very readable accounting of the Fatimid ideology and rule.

You might also check out the 15 Minute History podcast on the Fatimids.