Playing EU4, I notice I can easily convert people through missionaries which seem impossibly easy. In real-life, how do nations and empires convert large amounts of populations to their state religion after conquest, what methods they use to convince them to change their faith?

Hi,

Speaking about the spread of Islam in the Middle-East, there exists a widespread idea of the camel-riding left-handed Bedouin warrior who would give you the choice between Islam and death. This is not very historical, however:

Numbers of conversion After the destruction of the Umayyads (who actively discouraged conversion using the argument that Islam was an Arabic religion for Arabs), the rate of conversion increased dramatically. There are no exact numbers, but Richard Bulliet estimated that by the beginning of the 10th century 80 to 90 percent of the population of Iraq had converted to Islam. It seems likely that numbers in Egypt would have been similar, and the death of Coptic language as a spoken language in the 14th-15th century may also give a clue to this process. The process outside of urban areas was likely to have been much slower, as is attested by the continued existence of non-Muslims (albeit in relatively low numbers) in different areas of the Middle-East.

Methods of conversion
The most obvious method employed was to make conversion to Islam socially and economically attractive. Non-Muslims would pay a higher tax on their property as well as a poll tax, making conversion economically beneficial. Individuals may also convert in order to gain prestige or in order to rise in the ranks of society.

Outside of the dar al-’islām, in our case Central Asia, the Caucasus, the Indian subcontinent and south-east Asia, we find that from the 14th to 15th centuries the activities of Sufi orders start to play an important role. Examples of these orders would be the Yasawiyya in Central Asia, the Qadariyya in the Caucasus, and the Naqshbandiyya in the Indian subcontinen. Sufi saints were considered to be able to perform miracles and would enter into highly spectacular debates with representatives of other religions, which is described in various sources from the period (such as Ibn Battuta, I'll look up a nice citation from him).

This is not to say there were no periods of repression of non-Muslim groups; (the threat of) physical violence in some cases undoubtedly influenced the rate of conversion at various times and in various places. The massacre of the Jews in Granada might have enticed some of their co-religionists to either leave the area or to convert to Islam. There are also some indications that this occured on some Southeast-Asian islands where local sultans would wage war on non-Muslim tribes. This is outside my area of specialisation, so I'm a bit apprehensive about it though.

tl;dr: (mass) conversion to Islam happened because of socio-economic benefits, miracle workers, (rarely) violence

Preliminary sources:
Bulliet, R. Conversion to Islam in the Medieval Period, 1979.
Clayer, N. & A. Popovic, "Tarīqa" in Vol. 10 of The Encyclopaedia of Islam, New Edition, 2000.
Kennedy, H. Chapter 7 of The Prophet and the Age of the Caliphates, 2001.
–––––– Muslim Spain and Portugal, 1996.
Penelas, M. "Some remarks on conversion to Islam in al-Andalus", 2002.