What is the bypass capacitor at the emitter of a common-emitter amplifier for?
The negative feedback consists in the fact that when the output signal increases, the current flowing in the collector (and so the emitter) also increases. If the current on the emitter increases the voltage across the Re increases and the potential (Ve) of the emitter becomes higher. So the Vbe of the transistor will get lower : Vbe = Vb - Ve. A lower Vbe bring to a lower gain. 
That's the negative feedback.
If you put that capacitor to ground you will reduce this effect because at the operating frequency the capacitor will have a low impedance and so in parallel with Re will lower the whole emitter impedance, reducing the feedback effect.