How can you make a person shut up?
Turn your back on the person, and keep YOUR mouth shut. (s)he will get bored with you and go bug someone else.
I'd say it means the person saying it is not the brightest star in the sky. The "shut up" part is fairly self-explanatory, but the rest is a little obscure. It sounds like something Biff "make like a tree, and get out of here" Tannen from Back to the Future would say. (The joke there, of course, is that it's supposed to be "make like a tree and leaf/leave".)
He sounds rude, immature and aggressive. If he tells you to shut up now, can you imagine the stuff he would say to you if you two were to argue? When a person has feelings for someone and wants to impress them, they don't tell that person to shut up. This guy is a jerk---do not date him or even hang around him. You deserve to be with someone who enjoys spending time talking with…