Air goes into the intake. It goes back behind your motor, by the firewall, and down into the turbo. Turbo blows that air through your intercooler (also known as a charge air cooler) where it becomes "charged". At that point the air is cooler and more compressed, which is what helps create more power in your engine.
Right before this wonderful and magical air is about to enter your throttle body and later into your engine, there is a big Y pipe. One side of that Y pipe goes right into your throttle body, and then the cylinders. The other side of the Y pipe leads up and around the side (left looking from the front, on the passenger side) of your engine, up and back to your firewall where it ends in a little dongle that makes noise.
The noisepipe is unnecessary added volume to the charged side of your turbo system. Removing it will effectively help the system get to full boost quicker. In my honest opinion, the noisepipe is like installing a plastic hood scoop on an old dodge neon that doesn't lead anywhere. It's purely cosmetic. I actually frown upon VW for doing such a thing
If they wanted to create more noise, they should have put in a less restrictive intake system. Their marketing department refers to it as the "original hot hatch", but I really think they could be doing more in that department. Ahh, I digress.
I just dropped $750 on an APR tune this week, so I am going to have to put the brakes on for a little while before spending dough on new parts, but you better believe that my next modification will be removing the noisepipe, either with a tiguan pipe or a neuspeed or similar aftermarket one.
Keep in mind, the solutions mentioned above don't only replace the noisepipe, they replace all of the piping between the intercooler and the throttle body, so you're going to get improved flow above and beyond the tiguan pipe with an aftermarket part as well.