The pin is poked back and forth once through the paper. Two folds stiffen the sheet, so more air can be moved. Without the folds, only a small region of the paper near the pin flexes, and the sound will be weaker.
This grammophone is a curious item. Power is used only to run the motor. The arm is still the non-electric kind with heavy steel needles. When you use the arm instead of a sheet of paper, sound travels through the hollow arm down into the bottom box, and comes out through the four holes on the front.