There are three theories of what Southeast Asianists call "Indianization": the warrior theory, the trader theory, and the Brahmin theory.

The warrior theory is largely discredited, so we can brush over it somewhat. In short, it holds that the early Hindu and Buddhist states in Southeast Asia were essentially Indian colonies. If you want to hear more about this, let me know, and I'll elaborate, but virtually nobody buys it anymore.

Most contemporary historians hold to either the trader theory or the Brahmin theory. Both of these posit the spread of South Asian culture via peaceful / mercantile means, but disagree as to the agents of change.

The trader theory is self-explanatory: South Asian merchants were agents of cultural influence. This idea's appeal lies in how localized South Asian traditions are in Southeast Asia, and how different they are than in South Asia. Go to Bali, or Angkor Wat in Cambodia, and you'll see what I mean. There's Hinduism, sure, but it's absolutely not the same Hinduism you find in South Asia. However, the trader theory runs up against a wall when we consider that in the early "Indianized" states, Hinduism and Buddhism were elite ideologies and the majority of the people beneath the king and his/her court still practiced local traditions of spirit worship.

And this leads us to the most widely accepted theory, the Brahmin theory. According to the Brahmin theory, Southeast Asian kings invited South Asian Brahmins to their courts as ritual specialists. This theory explains why South Asian traditions remained elite and exclusive at first, and it's also a documented process - we can identify foreign Brahmins in early sources from Cambodia, and later sources from all around the region. It's a tradition that continues today - there are court Brahmins in modern Thailand, for example. It doesn't explain everything (aspects of the trader theory are certainly also true), but it's the most convincing explanation we currently have.

Recommended reading:

• Milton Osborne, Southeast Asia: An Introductory History (basic overview - read the first two chapters for this subject)
• Ian Mabbett and David Chandler, The Khmers (excellent pre-modern survey of Cambodia, lots of stuff on Indianization)
• George Coedes, the Indianized States of Southeast Asia (dated, but classic and very readable)

If you want any particular country studies, let me know, and I'll expand this list.