According to [John C Harrison](https://www.mnsu.edu/comdis/kuster/Infostuttering/Harrison/redefining-stuttering.pdf) (paraphrased in my own words): You have a subconscious belief that you are not allowed to speak your mind, that what you have to say is not important and that nobody would want to listen to you (or something similar and equally limiting). This causes you to "hold back" your speech (which causes or exacerbates your stuttering). If you speak in an accent (or you act out a role), then you are not really being yourself, so the shame that you have around speaking your own thoughts, is much less pronounced. That's why you are not inclined to "hold back" as much.