At this idiom the image that my mind comes up with is of the person who has been eating busily, heavily or carelessly and as a result has food/food crumbs around their lips. Don't know why (this image).
A mealy-mouthed person is someone who says unclearly what he means, usually in order to hide the truth, isn't it?
It's usually because the person is not confident enough to tell the truth or is afraid of being direct and to the point for some reason. It could have something to do with not wanting to reveal the truth - or wanting to hide the truth.