Essentially because of one man, Professor Michael Daniel Jones.
Jones was born in 1822 in a town called Bala, in Wales. The son of a Welsh Congregationalist minister, who in fact also ran a seminary, he grew up surrounded by relatively progressive non-conformist religion, and in a primarily Welsh-speaking environment. Predictably, he went into the family business as it were; and propelled by Victorian zeal, he emigrated to Cincinnati in 1848.
It's important to establish some wider context around this move. Although he recorded his own motivations as a desire to minister to other recent Welsh immigrants in the US, his sister had preceded him by several years, and the population of Bala had been dropping steadily for decades, as people moved away primarily to the rapidly industrializing South Wales valleys as well as overseas. This would seem to have reduced demand for his services in his home area.
In Ohio, he was disappointed to find not a coherent Welsh diaspora, but an immigrant population keen to integrate wholly into the local society. In particular, he lamented the fact that the Welsh language was not being spoken by the children of immigrants. He remained in Cincinnati for just two years, and returned to Wales somewhat depressed by the difficulty new arrivals had in finding work. Notably, he seems to have assumed that these difficulties were unique for Welsh immigrants; equally notably, he did not minister to or work with any other groups. (e: Wyn James characterizes this two years as 'a visit' but in those of Jones' journal extracts I have seen written prior to him traveling, he seems to be full of optimism for an entirely new life).
There are a couple of other 'push factors' in Wales at around this time. Wales had been the site of a lot of political unrest from the 1810s right up to the 1840s, with the Chartist uprising, Rebecca riots, and proto socialist 'beneficial societies' engaged in direct action in the workplace. Part of the British government's response to these problems of disorder was to conduct an inquiry into the state of education in Wales, which resulted in a report concluding that the Welsh language should, basically, be extinguished.
So this is the Wales to which Jones returned. He took over the running of his father's seminary, but was soon embroiled in a dispute concerning the language used to teach in his own institution.
It seems that an idea of a remote colony of Welsh speakers, designed to protect the language, took hold of Jones around this time. He began recruiting people for the project, and also researched possible sites for his project - including Palestine, bizarrely. The magic 'pull factor', however, only occurred when Argentina began offering generous land grants to European settlers in 1856. An exploratory journey in 1862 led to contracts being signed, and the first Welsh settlers arrived in 'Y Wladfa' (The Colony) in 1865.
Sources for the foregoing are Michael D. Jones a'i Wladfa Gymreig, edited by E. Wyn James and Bill Jones (2009) and Hanes Cymru by Ioan Davies (1994). I am not an historian, alas, but was born in Wales to a rigorously widely read family.
I know little but found this BBC report that goes on to this same subject.