"agree to disagree" is a nice way of saying "I've decided I won't try to kill you today"
That you've made the decision further energy is wasted trying to move someone's opinion...
Example: some relatives in my family have ideological positions unmoved by any evidence to the contrary. I can keep sliding evidence in front of them or move on to more productive activities. In a way, to me it says I no longer care so much about them - sadly, to them they think they've won the argument... by virtue of being obstinate without evidence.