There's actually a science behind this. It's to do with pathways in the brain, quite hard to explain, something like the dual-processing theory. Basically there's a subconscious route we use when speaking normally, and because we aren't really focusing on it, a stutter can appear. But when we are actively changing our speech/language to something different we are using it more consciously and the stutter is less likely to occur. Best example I can think of is with singing as the rhythm is different, like I said hard to explain but I hope it helps! I'm a SLT student and we were talking about this in our lectures a few weeks ago.