As I know barely enough to talk about this topic as it is I will stick mostly to Cambodia with some later references to Thailand but that will make sense as we go.

Well a relatively short answer to what is a pretty complex question would be a process known as “Indianization”, (which is refers to something along the lines of what it sounds like) that took place around 2000 or so years ago. The first aspect of this can be seen in the geography of the region, while the term is slightly unfashionable now, the region of Southeast Asia referred to as ‘Indochina’, denotes the area between the so called ‘great civilisations’ associated with India and China. Cambodia takes more cultural influence from the ‘Indo’ than the ‘China’ in this split. While the ancestors of the Thai people originated in Southern China I believe they migrated into the region that correlates with modern day Thailand at a later time (8th century perhaps) so I will get to them later in the answer. The Khmer, or the people of Cambodia have been there for thousands of years.

Historian David Chandler explains that a shared culture of ‘Monsoon Asia’, where the role of ancestral, tutelary deities is emphasised in the agricultural cycle, can be seen between India and Cambodia. For instance, a first century traveller from India would probably recognise similar rituals honouring the god Siva and the same would be said of someone travelling from Cambodia to India. In ‘A History of Cambodia’, Chandler says ‘During the first five hundred years or so of the current era, India provided Cambodia with a writing system, a pantheon, meters for poetry, a language (Sanskrit) to write in, a vocabulary of social hierarchies (not the same as the caste system), Buddhism, the idea of universal kingship, and new ways of looking at politics, sociology, architecture, iconography, astronomy, and aesthetics. Without India Angkor [the famous capital of the Khmer Empire] would never have been built, yet, Angkor was never an Indian city any more than medieval Paris was a Roman one.’

This process where certain Indian customs and practises came to influence Cambodian society was not imposed by force or colonisation, but I feel can be thought of as a kind of ‘osmosis’ where certain ideas were transmitted along the lines of trade routes where similar cultures found similar values amenable.

The reason that the same influences can be seen, although slightly less so, in Thai culture is that I would argue that a similar process that occurred between India and Cambodia in the 1st century occurred between the early Thai Kingdoms and the Khmer from the 10th century onward. While at the height of the Khmer Empire (before the 13th century) the early Thai people were subjects of Angkor. This dynamic changed as the Thais eventually rose up and defeated the Khmer, perhaps to an extent that Angkor was abandoned in favour of Phnom Penh.

The relationship between the two kingdoms was more complex than mere warfare however and there was a kind of cultural blending that occurred in this period after the Khmer Empire receded. Large transfers of people between both kingdoms facilitated shared customs and knowledge. So while the Indian traditions were initially transferred and perhaps ‘watered down’ in their transmission to Cambodia, these same traditions (now Khmer-ised) were again transferred as the Thai kingdom rose on what had been Khmer territory in the previous centuries. This is a slightly complex notion so I’m going to rely on again on Chandler. So here he is talking about the ‘newly established trading kingdoms of the Thai and Khmer and what he calls ‘the emulation factor’…

‘By the 1400s, Ayudhya (the thai capital) and these Cambodian cities looked to each other rather than to a Brahmanical past for exemplarly behaviour. Until the end of the sixteenth century, moreover, Phnom Penh (or Lovek or Udong {these were other Khmer capitals in the region} and Ayudhya considered themselves not separate polities but participants in a hybrid culture. The mixture contained elements of Hinduized kingship, traceable to Angkor, and Theravada monarchic accessibility traceable to the Mon Kingdoms (the Mon are a nearby group in what is now Burma) which had practised Theravada Buddhism for almost a thousand years, as well as remnants of paternalistic, village-oriented leadership traceable to the ethnic forerunners of the thai. Throughout the 14th and 15th century the official language common to both kingdoms was probably khmer. In Both societies the Buddhist sangha, or monastic order, was accessible in its lower reaches to ordinary people. Brought into contact with each other through wars, immigration, and a shared religion, the newly established thai and khmer kingdoms blended with each other and developed differently from their separate forebears.’

I hope that goes a little way into answering your question, which like I said is rather complex but can be answered fairly simply by looking at this early period of knowledge transfer from India to Cambodia in the ancient past and then through the relationship the Thai people had to the Khmer Empire in the medieval period.