Overall, the growth of Islam throughout the history of Islamic civilization was affected by social pressure (desire to increase status) and colonization.

It really depends which period and specific Islamic empire you are discussing, but there are some general patterns. The process of Islamification itself also changed over time with regard to how it happened in specific places. During the early Islamic period of the Rushidun Caliphate of the late seventh century and into the early medieval period, the Arab military class viewed their religion as being revealed only to them. This also reinforced their social order of an Arab military minority living in a garrison-town separate from their Christian, Jewish, or pagan subjects. Over time, however, religious scholars start to advocate a universal faith and people seeking to advance in status begin converting. In the second case, becoming Muslim was necessary to enter the elite during the initial period of Arab conquests. I say Arab instead of Muslim here because as Robert Hoyland points out in his book In God's Path the original period of expansion was driven by Arabs primarily. This also addresses the point about forced conversions: since Arab Muslims perceived their faith to be an ethno-religion, it did not make sense to impose conversion on their subject populations. However, later in the Islamic medieval era Turkic migrations and the waning of Arab power affirms that Islam is not just an Arab religion. Later on, the Ottoman Empire was very open to converts from European backgrounds, and many in the elite were ethnically Serbian, Albanian, Bosnian, and even Italian.

Colonization was another important factor in the spread of Islam. As I wrote above, the proliferation of these Arab garrison communities affected colonization in formerly Roman-Christian regions such as Egypt. Cairo as a city started as one of these military settlements. In places like medieval Islamic Spain there was a similar colonization following the invasion of Iberia. This continued in the early modern period with the Ottoman Empire who had a similar policy of settling Turkic peoples and Muslims in newly conquered areas to reinforce their control.

As for why Christianity declined areas where Islamic civilizations ruled, there were usually laws against the proselytizing to Muslims or conducting public worship such as processions or religious festivals. Under Ottoman rule for example, the Greek Orthodox Church was still allowed to exist but they could not build new churches. In other areas, like the Balkans or Central Europe where the Ottomans conquered, many Catholic bishops or cardinals would evacuate (as did many of their congregations) leaving the community still in Muslim territory without leadership. In these cases its easy to see where you would have traditionally Christian communities falling away or having difficulty maintaining their practice. Captured churches were also often turned into mosques as well. Officially, Christianity was tolerated as I mentioned earlier but people should be careful not to confuse that with "freedom of religion" in a modern sense. Christians and Jews were required to pay a jizya tax as non-Muslims throughout the history of Islamic civilization. Chattel slaves in the Ottoman world usually came from Christian backgrounds, for example slaves captured by the Crimean Tatars came from Russia or the Polish-Lithuanian Commonwealth. Other slaves came from Africa.