Traveller history is a highly contentious subject, because it is tied up with their identity and right to continue their itinerant lifestyle. The idea is that if Travellers are really just displaced or dispossessed people (as has been argued), the best "solution" is to re-settle them. This was the basic finding of the 1963 Commission on Itinerancy, which started under the assumption (in fairly unfortunate wording) that “there can be no final solution to the problem created by itinerants until they are absorbed into the general community."
Now, the real answer to this question is probably a variety of factors, with no one clear origin. But due to the contentiousness of the issue, certain "explanations" have become popular, which I'll outline:
One traditional view of Travellers is as displaced landowners from the Cromwellian invasions, and similar claims have been made that they are made up of people evicted during the Famine.
Many Travellers themselves point to a pre-Gaelic origin, asserting that they are the "original" inhabitants of the island.
Connections have been made to the Romani and other itinerant groups on the continent.
Travellers have also traditionally held professions that would have been conducive to being on the road, prompting a "chicken-and-egg" kind of debate (did they take up those jobs because they were on the road, or did they go on the road because of their jobs?).
The evidence for and against these claims is similarly complicated, but includes
The traditional Traveller language, known as Shelta, Gammon, or Cant, has many words of Romani origin, but is mainly influenced by Irish and English.
A 2011 genetic study found Travellers to be a distinct genetic population from settled Irish people, and claimed that the populations split up to 1,000 years ago.
The popularity of certain surnames such as McCarthy, which also show up among the Gaelic nobility driven out by Cromwell, has been used to suggest the displaced families theory.
The term "tinker," which describes the metalwork traditionally done by Travellers and has been used as both an endo- and exonym for the group, shows up as "tynker" or "tynkler" in the 12th century, although whether or not it was linked with a class of nomadic people is debated.
One issue that we have is that Travellers are not well distinguished from the general settled population in most literature before about the 19th century. Míceal Hayes mentions the issues surrounding the documented history (or lack thereof) of Travellers in his article "Indigenous Otherness: Some Aspects of Irish Traveller Social History," noting that
The social history of Travellers remained undocumented under colonialism, as successive British administrations did not distinguish between Travellers and the Irish poor generally. Consequently, there are real limits to what the sources can tell us about the history of the Irish Travellers. Those historical records that do exist simply highlight the relative invisibility of Travellers as a group in Irish society.
Really, the answer is probably a mix of all of the above. In every society, there are people who take to the road due to economic and social pressures, or just because they prefer the lifestyle. The popular song "O'Sullivan's John," written by the Pecker Dunne, tells the (supposedly true) story of a settled man who marries a Traveller woman and joins her. Presumably, whatever their origins, Travellers have picked up quite a few such people along the way, and it may be hard to find the "original" Travellers among all the others that have joined throughout the centuries. They have also mixed with other itinerant groups like the Romani, although this is more the case in Britain than it is in Ireland, where Romani populations are fairly low compared to the rest of Europe.
The "displaced persons" idea has been largely discredited, mainly because, while there were indeed many such people during the various invasions, wars, and famines throughout Irish history, there is little contemporary evidence that a significant group decided en masse to stay on the road indefinitely. Again, some of those displaced people probably were absorbed into the Traveller population, but that population seems to have existed long before Cromwell set foot on the island. Other, more fanciful ideas, such as Travellers being a lost tribe of Israelites, have been similarly rejected. The best we can say is that there were probably a combination of factors that have contributed to the Traveller community.
I'd like to plug the audio documentary I made as part of my masters dissertation on singing and music in the Traveller community, which deals with Traveller identity as they become increasingly settled. The accompanying written commentary includes a brief summary of some of the history, and also has a transcription in case the accents are too confusing (they get very thick!).