To me it's less about patience and more about urgency. Like when I say "please be patient with me" I more mean don't make me feel rushed. When I feel rushed to say something I'll definitely stammer more but when I don't feel rushed at all I stammer way less. So it's in both of our best interests for the listener to be patient with me. Plus I could give less of a F if someone is bothered by my stammer, it's their fault/problem if their bothered, not mine