The short answer is religion, proximity, and sympathy.

The American perspective of the IRA's existence and operation was obviously not uniform, and the majority of Americans remained apathetic, but there was a significant number of Americans who were sympathetic to the cause. These Americans were largely clustered in the northeast of the country, in cities such as Boston, where large quantities of Irish-Americans lived. At the time of The Troubles, the era of the Potato Famine and horrible anti-Irish racist sentiment were not far removed from the Irish-American consciousness. Additionally, Irish families would go to Catholic mass every Sunday and hear from the elders after service, where many were able to recall the brutal suppression of Irish freedom fighters in 1917. It was the religious and cultural ties to Ireland that made many sympathetic to the cause. Today it is estimated that 10% of the US population has Irish ancestry, which is a very significant figure.

But what about those who were not Irish and still supported the cause? These people fall mainly into two groups: socialists and patriots. The socialists would want to see a united Ireland to progress the values of socialism across the world, as having a country as significant as Ireland being socialist would bring further legitimacy to the somewhat experimental (at the time) ideology. The patriots fall into a much simpler sect of the American psyche. Although it may seem quaint, the US is a nation descended from freedom fighters, and many saw similarities between the Irish struggle and the American one. This was perpetuated by two larger factors: The British shedding her colonies after the Second World War; and the casual yet ever-present racism found in the United States. Some saw Ireland as just another imperialist colony that should be freed, and many saw Ireland subconsciously as a nation of white people who should inherently be free, just as the US was.

The final point is proximity. The US is an ocean removed from Ireland, and the activities of the Provisional Irish Republican Army and other more radical groups were published in the US, but their impact in terms of the lives they took was not explicitly condemned by the press. In Belfast a car bomb would tear through a community, hurting the lives of both Catholic and Protestant alike, but in the US, the bomb could easily be portrayed as just another act of war between two struggling sides. Tom Clancy's Patriot Games is actually very good at depicting this reality, as the official IRA wants nothing to do with the fictional ULA's terrorist activities in the United States for fear of ruining their image. (See, sometimes watching all those action movies can pay off!)

My answer was adapted from my two favorite books on The Troubles: Patrick Radden Keefe's Say Nothing, and Pringle and Jacobson's Those are Real Bullets.

Here's also a great article written by Kevin Cullen for PBS describing some more nuanced variables about the American reaction to the troubles, including the cognitive dissonance of being sympathetic to the IRA but also being an American conservative.

If I've broken any rules, I apologize, I don't often get the chance to post here!

It partly depends on which time period is being romanticized - the IRA of the 1920s was not the IRA of the '70s, though the latter iterations did much in their marketing to link themselves to it, and some similar tactics were utilized.

The truth of it is that not that many Americans supported the IRA. There are well over 40 million people who identify as being of Irish descent. The population of Ireland is still lower today (~5 million) than it was before the mass exodus of the mid-19th century (~8 million), when millions left for the US, Canada, Australia, Argentina, etc.

Many of those Irish communities in America stayed close-knit, often forming enclaves due to pricing of housing and having a connection to other Irish in the area. This was exacerbated by the general lack of hiring prospects and discrimination by WASP upper class Americans and hirers was a sour reflection of conditions they faced back home, though many laborer jobs were available. For example, the salt boilers of upstate NY or mines of Appalachia provided employment opportunities for men and older sons to send money home to larger cities along the coast, primarily NYC and Boston.

While the Easter Rising, Irish War for Independence, and the Irish Civil War did not result in large numbers of refugees, it did result in a number of well-documented atrocities by the British, such as the Burning of Cork.

Economic refugees continued to leave Ireland, though not in the numbers seen in the mid 19th century. They carried memories of British rule - the poverty that they saw as a result of British rule, the actual war with Britain, etc. My great great grandparents came over from Ireland during the 1910s, and my great grandmother was an infant or toddler at the time. Yet she had enough ingrained and cultural carrots and sticks growing up in an Irish immigrant community that she attempted to donate money to the IRA as an adult.

So these stories become part of family histories. If you grow up hearing about British auxiliaries burning down your great uncle's shop, you have a degree of ingrained beliefs about the situation over there.

The later iterations of the IRA played to this history - getting Republican leaning Irish bands playing their songs that glorify the revolts and deeds of Irish against the English going all the way back to Cromwell is a form of legitimizing the movement and raising funds and awareness that the British could not compete with. Even recently the BBC barred Rod Stewart from performing two songs due their anti-English nature.

The British actions did little to help matters - the Bloody Sunday shooting of protestors in 1972 was fodder for direct comparisons to Kent State or even the Boston Massacre. So it was easy for the whole affair to be sold to the Americans as a bunch of underdogs fighting against a colonial oppressor that had to resort to violence. It fits very well with the US national mythos - helped by the fact that well over a hundred thousand Irish immigrants served the Union in the US Civil War.

The Anglo-Irish Agreement was even crafted by Boston-area Irish-descended American politicians, further solidifying this connection.

TL;DR: The narrative that Americans hear is typically the Irish one, the British committed acts that remind us Yankees of both the worst of the Vietnam/Civil Rights protest massacres, and the conflict has echoes of the US independence war. Also music and family histories have done a great job linking pthe IRA as the successor to past revolts against the British/English.

Sources: (sorry if the formatting is inconsistent)

Frontline, PBS/WNYC: America and the conflict: https://www.pbs.org/wgbh/pages/frontline/shows/ira/reports/america.html by Kevin Cullen

The Irish Times: "Anglo-Irish Agreement 'a great shock' to Unionists, FitzGerald told Reagan" by Elaine Edwards. Friday, 30 Dec 2016.

Washington Post "Why a 1972 Northern Ireland murder matters so much to historians" by Donald Beaudette and Laura Weinstein, 6 Nov 2019.

"No 'Wild Geese' this time"? IRA Emigration after the Irish Civil War by Gavin Foster. Irish-American Cultural Institute vol 47, Issue 1&2, Spring/Summer 2012 (pg 94-122).

"They banned one of Ireland's greatest ballad - what was the BBC thinking?" Kathleen Maloney 01 Oct 2018. Irish Central. https://www.irishcentral.com/opinion/others/bbc-rod-stewart-grace