It means don't start giving me trouble, especially verbal abuse. You walk in the house two hours late for dinner, you were late because of traffic, and then you had a flat tire. Your wife starts to reprimand you for being late. You hold up your hand and say, "don't start with me!".

Hi Roby, You would say it when people want to start a fight with you and you want to scare them. Or imagine a situation when you are angry and somone wants to start a fight with you or asks you something that annoys you, then you will say 'Don't start with me!' as a sign so they leave you alone.