An emotional cut-off is when an individual or multiple family members decide to emotionally distance themselves from others.
Cut-offs happen when there is a lot of tension and anxiety, and hurt feelings. The person doing the cutting off feels that the best way or only way they can deal with these unpleasant feelings is to just cut the person out of their life emotionally and physically by no longer interacting with that person or those family members.
This often happens when a family has a strong belief or value, and an individual is seen as violating that belief or value.
Suppose a family is anti-homosexual and a family member reveals that they are homosexual. In that case, the family may have strong conflicting emotions and decide that the best way to resolve their conflict is to cut the person off.
These cut-offs can happen for many reasons, small or big; all that matters is that an individual or group doesn't know how to deal with their upsetting emotions. The problem with using emotional cut-off for relief is that it's a temporary fix, not a resolution.