They just want the (other person's) feelings to go away.
While it may seem that this is the case with conflict/intimacy avoiders in many ways this is just a result of a fear of their own emotions . Even more than anothers emotions their own strong feelings are trying to be negated, they are afraid of their own feelings. I used to be like this avoiding conflict because I did not know how to deal with the anger that would arise inside of myself.