How did the Shia dynasty of the Fatimid come to control the Sunni kingdom of Egypt ?

Beware long post that touches on Shia schisms and how they forced the proto Fatimids into North Africa I'll split it in a reply to my initial post:

To answer this question it's important to understand how Ismaili belief spread and fractured in the pre Fatimid period.

Background:

The Ismaili schism:

As you noted the Fatimids were Shia, however they were Ismaili Shia and not Ithna Ashari Shia that is more common today. To simplify a little bit the last Imam that Ismailis and Ithna Asharis have in common is Jafar as Sadiq. He designated his son Ismail to succeed him after his death, however it is generally believed that Ismail predeceased his father. The Jafari Shia split because of this: a large number following Ismail's full brother Abdullah al-Aftah becoming the Fathiyya, another group following his half brother Musa al-Kadhim becoming what would eventually become the Ithna Asharis, another group following Ismail's lineage and therefore his oldest son Muhammad ibn Ismail and another believing that Ismail had faked his death as he was rumoured to be involved in anti Abbasid plots and had gone into hiding.

Both of the last two groups mentioned are Ismaili and their different beliefs about how the role of Imam was passed leads to conflict further down the road. Herersiographers termed the first group the Mubarakiyya and the second group the pure Ismailis (al-Ismailiyya al-khalisa). The Mubarakiyya later split again on Muhammad ibn Ismail's death with most believing he did not die but went into occultation (spiritual hiding) and would return as the Mahdi (Messiah), and the rest following his lineage.

The Ismaili Shia opposed the Abbasid Caliphate with dais (missionaries or more literally summoners/callers) converting people around Iraq, Iran, Yemen, and Eastern Arabia. This obviously put them at odds with the Abbasids and their supporters. However it was very appealing to the Ithna Asharis (although this occurred before the occultation of the 12th Imam of the Ithna Asharis and much it before his birth so the name Ithna Ashari may not be entirely correct) who were unsatisfied with their own Imams keeping their heads down and avoiding opposition to the Abbasids. This led to a large number of conversions to both Ismailism and Zaidism, which is another Shia sect.

How the proto Fatimids reached North Africa:

Ismaili belief in Iraq was largely spread by Hamdan Qarmat and those loyal to him. There was a sizable Ismaili population in Iraq as nearly all Ismailis believed that the Imam had fled from Medina to Iraq to hide from the Abbasids (Basrah for Ismail and Kufa for Muhammad) but Qarmat was incredibly successful in spreading it further. Qarmat was so successful in converting the populace that Ismailism was termed Qaramita in Iraq by those unfamiliar with Ismaili belief and the term was even used for those Ismailis not led by Qarmat and separate from him in everything but Ismaili belief. The Ismailis referred to themselves as al-dawa (the mission) and al-dawa al-hadiya (the rightly guiding mission).

Qarmat sent Abu Said al-Jannabi to Bahrain where he was successful in converting the local populace and from there Ismailism spread to Yemen.

From Yemen Abu Abd Allah al-Shi'i was sent to the Maghrib and converted the Kutama Berbers to Ismailism (although there's also the claim that al-Shi'i met with Kutama leaders in Mecca and accompanied them back to the Maghrib were he converted the population).

In 899 the future Fatimid Caliph Abdullah succeeded to the central leadership of the Ismaili mission now based in Salamiyya, Syria, and claimed the Imamate for himself and by extension his deceased father, grandfather and great grandfather. He instructed Qarmat and the other chief dais to start preaching his Imamate rather than the Mahdiship of Muhammad ibn Ismail. Qarmat refused. This split the Ismaili community with the Ismailis loyal to Abdullah centred largely in Yemen and North Africa and those who continued the belief in the Mahdiship of Muhammad ibn Ismail based largely East of Egypt and across the Arabian peninsula and Iran.

Abu Said al-Jannabi, a dai loyal to Qarmat set up a state in Bahrain in the same year. It's here were the term Qaramita and more (in)famously Qarmatian begins to apply to those Ismailis opposed to anyone claiming the Imamate after Muhammad ibn Ismail.

Resentment and hostility grew between the Qaramita and proto Fatimids over the next few years and in 902 Abdullah was forced to flee to North Africa as the Kutama Berbers mentioned earlier were firmly in support of Abdullah's claim to the Imamate; likely due to al-Shi'i's support of his claim.

Whilst in North Africa Abdullah and al-Shi'i were able to raise a force of primarily Kutama Berbers to overthrow the Aghlabids who were weakened by war with the Tulunids, and a Berber uprising (including some Kutama but not limited to just them). In March of 909 the Ismailis took the former Aghlabid capital of Raqqada, and Abdullah entered the city in January 910 and was proclaimed Caliph. Here is the birth of the Fatimid Caliphate.

Answering the question: The Fatimids Conquest of Egypt:

The defeat of the Aghlabids put the Fatimids in opposition to the Abbasids militarily rather than as dissidents from within the Abbasid Caliphate.

The Fatimids attempted to take Egypt on several occasions, before eventually succeeding. The first attempt in 914 took several important cities most notably Alexandria before being pushed back by Abbasid reinforcements. The Fatimids did maintain control of Barqa in modern Libya from which they could stage later invasions.

Another attempted invasion in 919 had much the same result.

In the late 920s to early 930s pretty much everything was going wrong for the Abbasids. Al-Muqtadir and al-Qahir swap the position of Abbasid Caliph a couple of times, the treasury is nearly gone and the Qarmatians(the same ones from earlier) sack Mecca and Medina during the Hajj and steal the black stone from the Kaaba. So the Fatimids try again, again they were repulsed but this time by the Ikhshidids, it seemed like the Ikhshidids were willing to cooperate with the Fatimids against the Abbasids until the Abbasids recognised the Ikhshidid Titles in Egypt, making them nominally an Abbasid Vassal but in real terms independent.

Abbasid power continued to weaken with conflict with the Buyids rendering them essentially a puppet state.

The Ikhshidids weren't doing any better suffering famine, raids, anti Christian riots after Ikhshidid defeat to the Eastern Roman (Byzantine) Navy, and a freed slave and eunuch called Kafur seizing power all led to a much weakened and divided state. In 968 Kufur died, being a eunuch he left no heir and various groups begin infighting.

By contrast the Fatimids were doing fantastically well, pushing the Umayyads out of much of Algeria and Morroco and had taken control of Sicily with the Byzantines agreeing to a truce in 967.

In 969 then Fatimid Caliph al-Muizz begins his invasion. Most of the Ikhshidid elites opted to surrender without fighting and some even welcomed the Fatimids as a stabilising influence. Alexandria was occupied without resistance. The Ikhshidids in Fustat, the then Egyptian Capital, agree to surrender to the Fatimids if the Fatimids don't prevent them fighting the Byzantines. In short the Ikhshidid soldiery refused the terms of their surrender and so fighting was inevitable. Fustat fell to the Fatimids in five days.

With the fall of Fustat the Fatimids had control of Egypt and could then turn their attention to putting down rebellions.

In celebration of their victory the Fatimids built a new capital, al-Qahirah (the Conqueror/the Victorious) which is Cairo today. It would be the seat of the Fatimid Ismaili Imamate until the end of the Fatimid Caliph Imams.