Derry isn't in the Republic of Ireland. It is in Northern Ireland, which is still part of the United Kingdom. From the late 1960s until 1998 there was a civil war in Northern Ireland, between Protestant Unionists who wanted to stay in the UK, and Catholic Republicans who wanted leave the UK and join Ireland. The British Army operated in Northern Ireland, ostensibly to prevent violence, but in reality they frequently cooperated with the Unionist militants. The men marching in orange were members of a Protestant fraternal organization, probably the Orange Order. Between April and August every year these Orangemen march to celebrate various Unionist events. Frequently they march through Catholic neighborhoods. During "the Troubles," these marches were flashpoint for serious violence. That is why the family in the show are leaving town. They are going to Ireland to avoid violence. And the guy they find in the trunk of their car is a Republican militant.

u/DJLookinAss answered a related question a few months ago, and included a lot of great sources. You should find his answeredifying.