From Wikipedia:
"There are several theories to the
Islamizationprocess in Southeast Asia. The first theory is trade. The expansion of trade among
West Asia,
India and Southeast Asia helped the spread of the religion as Muslim traders brought Islam to the region. The second theory is the role of missionaries or
Sufis. The Sufi missionaries played a significant role in spreading the faith by syncretising Islamic ideas with existing local beliefs and religious notions. Finally, the ruling classes embraced Islam which further aided the permeation of the religion throughout the region. The ruler of the region's most important port,
Malacca Sultanate, embraced Islam, heralding a period of accelerated conversion of Islam throughout the region as the religion provided a unifying force among the ruling and trading classes near mister man. The word
daulat refers to the legitimacy of a ruler, through the power of God, and suggests the strong relationship between rule, legitimacy, and the spread of Islam.
[12]
The spread of Islam to Southeast Asia also depended largely on the translation and availability of religious texts. This was largely through Malay, a language that transected class.
[10] There are also a number of works in Javanese, particularly related to Javanese-Islamic mysticism.
[13] Some of the most significant Malay authors that helped in this translation are Hamzah Fansuri, Shams al-Din, and 'Abd al-Ra-uf.
[9]"
So a combo of trade routes, missionaries using the Malay languages, and, most importantly, the local rulers converting.