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The basic issue is that a central part of the 1998 Good Friday Agreement that ended the seemingly endemic ethno-religious-political violence in Northern Ireland was that both Northern Ireland and Ireland were part of the EU.
This meant they effectively had no border controls, so Irish Nationalists in Northern Ireland could consider their political separation a mere administrative matter, maintain Irish citizenship in many cases, and still picture themselves as part of a political unit with the rest of Ireland.
This is why the more Irish sections of Northern Ireland voted the most strongly against Brexit in the first place.
Both countries being in a borderless EU was essentially an inherent condition of the peace agreement. So the UK leaving the EU and taking Northern Ireland with them threatens to destroy the entire agreement.
Brexit essentially represents the British side unilaterally breaking the peace agreement.