Many famous supporters of Irish republicanism have been Protestant: Wolfe Tone, Henry Grattan, Robert Emmet, and Charles Parnell, to name a few. How did this struggle become (by the time of The Troubles) an almost purely sectarian, Catholic vs. Protestant fight?
The divisions between Catholics and Protestants in Ireland have deep roots. Anti-Catholicism in England really hit a fever pitch by James I's reign (1603-1625) due to the Gunpowder Treason and Plot among other things. This is important because when James I began his plantation scheme in Ulster, he used Protestants who were viewed as loyal. The plantation scheme also took Catholic lands and gave them to these Protestants which led to multiple violent episodes between people of the two faiths (and yes, both sides were guilty of doing terrible things). Eventually, 17th century rebellions and the Glorious Revolution added up to the Penal Codes, which stripped Catholics and Dissenters of their rights. Quakers and Presbyterians in particular were sympathetic to Catholics during that time because they also had some disabilities under the law. The few Anglicans who sympathized were mostly driven by Enlightenment rhetoric that stressed equality for all men irrespective of faith.
This is where men like Wolfe Tone come in for the 1798 Rebellion and Emmet's Rising. There's a broad historiographic consensus that views the 1798 Rebellion as a missed opportunity during which the "two Irelands" could have worked together.
Catholic Emancipation in 1829 scared Anglicans a great deal because now Catholics could hold office and they were organizing in great numbers. The biggest divisions came later in 1886, when the first Home Rule Bill (Government of Ireland Bill) became a hotly debated topic. At this point, many Protestants began to fear that Home Rule (an Irish Parliament) would equal "Rome Rule". This is roughly where modern sense of Irish sectarianism begins. Tensions began to boil over and there was a lot of rioting in mixed areas like Belfast between 1886 and the 1920s. Anti-Catholicism was always simmering just beneath the surface, all they needed was a context for violence and division to rear their ugly heads.
The majority of Irish people who were against Home Rule were Protestant, although there were a handful that did support Home Rule. Ultimately, Home Rule was finally passed in 1914, but suspended due to WWI. The Protestants felt betrayed by the British government's support for Home Rule and threatened civil war; they formed the Ulster Volunteer Force in January 1913 for that purpose. The nationalists were predominantly Catholic, but did have a few prominent Protestants among them, such as Constance Markievicz and W. B. Yeats. The Irish Volunteers/IRA were founded in November 1913 to fight back against the emerging Protestant paramilitarism. The country became increasingly divided after the 1916 Rising, which was led by the Irish Volunteers and Irish Citizen Army.
The final issue was the implementation of Home Rule. After WWI and the war in Ireland ended, the new Government of Ireland Bills established two new states: The Irish Free State and Northern Ireland. The nationalists (mostly Catholic) were outraged because they wanted one nation and the unionists (mostly Protestant) feared their neighbors so much that they oppressed the Catholics in the north in a multitude of ways. Gerrymandering votes, discrimination against Catholics in housing, and a whole host of other problems caused the most extreme divide between Catholics and Protestants in Northern Ireland.
The period known as "The Troubles" actually began much later (the 1960s) within Northern Ireland. It came about from a largely peaceful Civil Rights movement that was predominantly led by Catholics. Again, mass meetings of Catholics contributed the Protestants' siege mentality and many of the Civil Rights marches ended with beatings and brutal police action (most famous example: Bloody Sunday).
Sources
Bardon, Jonathan. A History of Ulster. Dundonald, Belfast, Northern Ireland: Blackstaff Press, 1992.
Bew, Paul. Ideology and the Irish Question: Ulster Unionism and Irish Nationalism, 1912-1916. Oxford: Clarendon Press, 1994.
_. Ireland The Politics of Enmity, 1789-2006. Oxford: Oxford University Press, 2007.
Bew, Paul, Peter Gibbon, and Henry Patterson. Northern Ireland 1921-2001: Political Forces and Social Classes. London: Serif, 2002.
Elliot, Marianne. Wolfe Tone: prophet of Irish Independence. New Haven: Yale University Press, 1989.