The paper cone is fixed to the frame of the speaker by a flexible section. As the current goes through the coil round the base of the speaker cone the cone is pulled either backwards or forwards. (See Force on a current in a magnetic field and Fleming's left hand rule). The bigger the current the more it moves, that is why you can see it in the large disco speakers when the sounds are large!
If the current changes, as it will from a sound source such as a microphone the cone is pulled backwards and forwards. The low notes will be of a lower frequency and so the cone will move backwards and forwards more slowly and so the movement is easier to see.