This topic has been the source of a lot of controversy, esp. since Quaritch Wales and a number of Indian scholars opened the discussion by explaining it as a case of Indian kings making conquests abroad (hence they referred to Hindu S.E. Asia as "Greater India" -- see the Journal of the Greater India Society, published from Calcutta from 1934).
In fact the most likely scenario: Indian traders seem to have operated in the eastern Indian Ocean beginning in the last couple of centuries BCE. Eventually, that trade came to be dominated by south Indians, who established trading colonies along the coasts. Local rulers, consolidating realms (based usually in in-land capitals), adopted Indic cultural styles (including writing) and religious beliefs and practices learned from the traders, eventually establishing contacts with Indian rulers, and inviting brahmins and monks to settle there to establish "classical" royal cults.
Over time, local ancestor cults and other religious beliefs were integrated into the local versions of the Indic traditions. Earlier claims that these features remained a veneer associated only with the royal courts are belied by the persistence of Indic names at many social levels, and of forms of Hinduism itself among remote (i.e., not subsequently Islamized) groups in Indonesia.
For an overview of what I take to be the current scholarly consensus on how Indian culture and religions spread to Southeast Asia, I recommend:
Kulke, Hermann, " India's Impact on Southeast Asia: Causes and Consequences," pp. 152-161 in:
Kulke, Hermann, and Dietmar Rothermund. (1990) A History of India. London: Routledge.
PRE-MODERN PERIODS, S.E. ASIA IN GENERAL
Bhattacharya, Kamaleswar (1955) "La secte des Paçupata dans l'ancien Cambodge." Journal asiatique 243: 479-490.
---- (1961) Les religions brahmaniques dans l'ancien Cambodge d'après l'épigraphie et l'iconographie. Paris: EFEO.
de Casparis, J. G. (1996) "Some Notes on Ancient Indian Ritual in Indonesia." in Ritual, State and History in South Asia: Essays in Honour of J. C. Heesterman, ed. by A. W. van den Hoek, D. H. A. Kolff, and M. S. Oort, pp. 480-492. Leiden: Brill. (de Casparis has written extensively on inscriptions.)
Christie, Anthony (1964) "The Political Use of Imported Religion: An Historical Example from Java." Archives de sociologie des religions 9(17): 53-62.
---- (1983) "Raja and Rama: The Classical State in Early Java." In: Centers, Symbols, and Hierarchies: Essays on the Classical States of Southeast Asia, Monograph Series, no. 26, Southeast Asia Studies. New Haven: Yale U.
Coedès, George (1932) "Études cambodgiennes, XXVIII-XXX." Bulletin de l'École française d'Extrême-Orient 32: 71-112.
---- (1937-1966) Les inscriptions du Cambodge, 8 vols. Paris: Editions de Boccard.
---- (1939) "La plus ancienne inscription en langue chame." New Indian Antiquary, extra ser. I: 46-49.
---- (1968) The Indianized States of Southeast Asia, edited by Walter F. Vella, translated by Susan B. Cowing. Honolulu: University of Hawaii Press.
Finot, Louis (1902) "Notes d'épigraphie, I: Deux nouvelles inscriptions de Bhadravarman Ier, roi de Champa." Bulletin de l'École française d'Extrême-Orient 2: 185-187.
---- (1918) "L'inscription de Chiêm-son." Bulletin de l'École française d'Extrême-Orient 18: 13-14.
Glover, Ian (1983) "Archaeological Evidence for Early Trade between India and South East Asia." In: The Indian Ocean in Antiquity, edited by Julian Reade, pp. 365-400. London: Kegan Paul.
Gomez, Luis O., and Hiram W. Woodward, Jr. (1981) Barabudur: History and Significance of a Buddhist Monument. Berkeley Buddhist Studies Series. Berkeley: Asian Humanities Press.
Gonda, Jan (1973) Sanskrit in Indonesia, 2d ed. New Delhi, International Academy of Indian Culture. (Orig. ed. 1952.)
---- (1975) "Siva in Indonesien," repr. in Selected Studies, vol. 4, pp. 91-121. Leiden: E. J. Brill.
---- (1975) "The Old-Javanese Agastyaparvan," repr. in Selected Studies, vol. 4, pp. 523-end. Leiden: E. J. Brill.
---- (1975) Selected Studies, vol. 5: Indonesian Linguistics. Leiden: Brill. (incl.: "The Javanese version of the Bhagavadgita" and "Einige Mitteilungen ueber das altjavanische Brahmanda Purana.")
Higham, Charles (1989) The Archaeology of Mainland Southeast Asia from 10,000 B.C. to the Fall of Angkor. Cambridge: Cambridge University Press.
---- (2001) The Civilization of Angkor. London: Weidenfeld & Nicolson.
Houben, Jan E. M., ed. (1996) Ideology and Status of Sanskrit. Leiden: Brill.
Kern, Hendrik (1917) Verspreide Geschriften, vol. 7.
Kulke, Hermann (1978) The Devaraja Cult, translated by I.W. Mabbett, with an introduction by the author and notes on the translation of Khmer terms by J.M. Jacob. Data Paper no. 108, Southeast Asia Program. Ithaca, N.Y.: Cornell University. Orig. pub.: (1974) "Der Devaraja-Kult." Saeculum 25(1): 24-55.
---- (1993) Kings and Cults: State Formation and Legitimation in India and Southeast Asia. New Delhi: Manohar.