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 was 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 set 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 perpetrated 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!)
This 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.