I don't know what to do to stop or curb it. My son does this sometimes.
He has a friend who literally says nearly every thing he thinks. He's a smart, sweet kid, so it doesn't seem odd at all until you've been focusing on what he's saying for a long time. I actually didn't notice it until his mom mentioned it; she called it, I think "a non-verbal speech disorder", meaning he has full range of speech ability but poor control in keeping his thoughts to himself.
I think this steady stream of chatter is fairly typical for kids on the spectrum though I never saw this mentioned as a symptom or even as a usual behavior for Asperger's or autism. There is a lot that isn't covered in all the books.
Do you feel like your son is getting enough sensory stimulation? I've found that when my son isn't getting those needs met he's more likely to self stim or display some destructive or disruptive behavior. Like he needs a set amount of friction in his life. If he can get it by jumping, climbing, or being allowed to wrap a blanket really tight around himself then he doesn't have to do those things that freak out strangers.