Hi I realise this answer comes a bit late, and it's gonna be short but I hope it helps.

The Fenian Brotherhood derived its name from the mythical Fianna, a group of warriors that followed Fionn Mac Cumhaill (Finn McCool) in a cycle of Irish mythology known as the Fenian cycle. Now there's a bit of a translation problem* here due to how languages evolve but essentially a fian was a band of warriors and a fianna was a group of bands with individual members being Fénnid/Feinne. Fénnid/Feinne meant warrior or soldier, as such the Fenian Brotherhood was a group of warriors/soldiers or a brotherhood if you will.

So how did the term for warrior evolve to become a slur? Simple, we've seen it happen in the past few years with another term, Jihadi, terrorism stained the word. The Fenian Dynamite campaign can really be seen as the birth of modern terrorism, as in the attacks were actually orchestrated with the intent to cause terror. Bombs, stabbings, assassination and even just plain old street violence brought the Irish question home to the British public in a way never before seen, newspapers ran stories of Fenian attacks, of perceived Fenian threats, of the Irish problem. Soon the ugly face of racism and sectarianism began to turn and the Irish community living in Great Britain felt the ire of a nation put on edge by sensationalism and racist depictions of the Irish as brutish and uncivilised saw the term fenian be applied to any and all, a way of calling them a terrorist.

Northern Ireland has a long history with sectarianism and there's a whole range of slurs to be thrown around, some interesting like fenian or taig, some sorta boring like hun or paddy and some down right confusing to outsiders like 'black', which has nothing to do with race. Recent history and conflict in NI has kept the old slur in use for much of the same reasons as it came about but also due to its ironic and reclaimed use by the Catholic Irish, badges carrying the phrase "still an unrepentant Fenian bastard" are sold by Sinn Féin and you even get it on a shirt.

Now I've skipped over a lot of complicated history here, largely due to the fact that it's irrelevant to why the word became a slur but I'll tl;dr it further here.

The Fenian Brotherhood carried out terrorist acts in the name of Irish nationalism, the term fenian was used to denote their members and their ideology. Newspapers sensationalised the stories with racist depictions of Irish people and eventually the word became a way to infer an Irish person was a terrorist. The word stayed in use despite the Fenian Brotherhood being replaced largely due to continued sectarianism in Northern Ireland and beyond.