In South East Asia in general the short answer is: through trade. There is even evidence that Islam first appeared in Indonesia in the 8th century.

Long answer: the decline of Hindu powers in India led to the increasing prominence of Muslim Indian and Persian traders. The 15th century saw civil wars in Majapahit's center while the Sultanate of Malacca grew increasingly powerful with Chinese support, so that through most of the century Majapahit's rule was basically just nominal and by the 16th century they were just a small Hindu principality, soon finished off by a coalition of Javanese Muslim states while the old Hindu aristocracy fled to Bali. To understand why Islam came to South East Asia, you have to see what SEA nations are Muslim today and what kind of people live there. So the people who are Muslim are Malaysian, Indonesian, and the Southern Filipinos (they consider themselves to be Malay). All of these people fall under the Malay culture group. They all live on islands with a ton of spices and exotic goods and coasts so their main way to sustain an economy has always been historically trading. Malay merchants would head west towards India and then Persia and Arabia with goods and come back richer. Some of them converted to Islam to gain some favorable trade and market relations with the Islamic merchants abroad. These converts brought the religion back to the Malay islands and started to proselytize the locals. The markets would be dominated by these Malay Muslim traders and shop owners and locals would again convert to get better deals, or if they wanted to get into ship trading they'd convert to get better business from the Middle East.

Eventually so much of the Malay had converted to the point that rulers would convert and style themselves as "Sultans". This was to legitimize themselves and better control/rule the populace. Then once you got the ruler to convert that was it. He would essentially force convert the remaining populace to consolidate his rule or the religious excuse of "expanding the Dar al Islam"

The nations of Malaysia and Indonesia were built on this. Then they got colonized by the British and Dutch, respectively. Brits and Dutch were not interested in converting, but extracting the economic wealth of these lands so the Islamification stayed.

Basically:

1. Wealthy Muslim merchants flocked around the spice trade routes, spreading Islam through marrying locals and settling down (a lot of Indonesians now still has surnames of Persian origin). Due to dominance of foreign Muslim merchants, local traders began converting to Islam in order to gain more favor with these foreigners. Eventually the wealthiest of these merchants would marry into the ruling elite of the Hindu-Buddhist kingdoms, thus establishing a future Islamic nobility (the first sultan of Demak was the son of one of the last Majapahit kings).
2. Zheng He's expedition (who was a Muslim) established some Chinese muslim settlements throughout the archipelago
3. At first the spread was gradual, until the emergence of powerful Muslim states (Malacca, Aceh, Demak, etc) led to the eventual dominance of Islam in the region.

What made conversions easier is the fact that they mixed Sunni Islam with Hinduism so for the first few centuries Indonesians and Malays practiced a hybrid Sunni-Hindu religion, still doing Hindu rituals and worshipping Hindu gods until the 19th and 20th centuries. Malays used to practice Hindu Mandi Safar bath until the 70's and Indonesian Abangans in Yogyakarta worship a Hindu goddess today. Indonesian Muslims used the names of Hindu gods like Wisnu (Vishnu) to name their children.