The addition of the emitter resistor means that the transistor’s emitter terminal is no longer grounded or at zero volt potential but sits at a small potential above it given by the Ohms Law equation of: VE = IE x RE. Where: IE is the actual emitter current.
Now if the supply voltage Vcc increases, the transistors collector current Ic also increases for a given load resistance. If the collector current increases, the corresponding emitter current must also increase causing the voltage drop across RE to increase, causing an increase in base voltage because VB = VE + VBE.
Since the base is held constant, the DC voltage on the base relative to the emitter Vbe is lowered thus reducing the base current and keeping the collector current from increasing. A similar action occurs if the supply voltage and collector current try to decrease.
In other words, the addition of this emitter resistance helps control the transistors base bias using negative feedback, which negates any attempted change in collector current with an opposing change in the base bias voltage and so the circuit tends to be stabilised at a fixed l