I think the statement reflects that sometimes you can be passing on a message, but it isn't the one coming from your mouth. When we were kids, my mother used to say, "What you are speaks so loud I can't hear what you're saying." It is easy to say something, but sometimes our "actions speak louder than words."
If someone tells us something we don't want to hear and/or don't agree with, rolling our eyes, crossing our arms, avoiding eye contact, snorting, or smirking are things that say much more than our words. Insincerity plays a part in this quote. If you're actions are not contradicting what you say, it's possible that your tone may be misleading. Sometimes people read things into our words that we do not intend. It's difficult to avoid this unless we are especially explicit in what we say. Other times we may speak and our tone does not seem genuine: if we are sarcastic or our responses lack conviction, what we are saying may not be heard at all.
In making sure we are understood while speaking, it is important to strive for clarity, to make sure our facial expressions match what we are saying (or want to say) and that we look for non-verbal clues from our listeners: a person we may be speaking to or a group. Asking for reinforcement like, "Does that make sense?" or "What do you think?" will provide you with instant feedback.
Miscommunication can destroy business deals, peace negotiations and friendships. It can be a tricky business. Always speak carefully and watch for the reactions of others if you're concerned about the effect of your words.