If you are trying to bias the transistor within a specific range of collector currents (regardless of which sample of the type of transistor drops out of the tape, regardless of resistor tolerances, supply tolerances and temperature) you will have a problem meeting that requirement without Re.
You could replace one of the resistors with a trimpot, but if Re = 0, the adjustment will be very fussy and when the transistor warms or cools the collector current will change a lot. As designers, we would like to avoid adjustments or special selection of components where possible, and have the circuit bias work well over a wide range of conditions.
Without Re, the bias current will be very dependent on Vbe and/or hFE**, transistor characteristics that are not tightly specified and which change with temperature. By adding Re, you can make the bias current dependent on Vbe+Vre, and if you make Vee = ReIe ~= ReIc >> Vbe, then the importance of Vbe is greatly reduced. By keeping R1||R2 low enough, the importance of hFE is reduced (too low and it wastes power).
** There is a degree of freedom with the voltage divider. If the resistors are very high value then the value of R1 and the transistor hFE characteristic dominates the value of bias current. In the extreme, R2 is open and R1 + hFE determines the collector current. If the resistors are very low value, the bias current variation is defined more by the relationship between Vbe and collector current. Vbe at a given collector current has a negative temperature coefficient. hFE at a given collector current usually has a positive temperature coefficent so they tend to add (collector current will increase with temperature).
R1/2 fix a voltage on the base, which fixes a voltage on the emitter.
With a voltage fixed on the emitter, a known current flows through RE, through the emitter, and almost all of it in the collector.
Normally, a circuit like this would have a resistor in the collector lead as well, for this current to generate a voltage across.
If the collector resistor was bigger than the emitter resistor, then the circuit would have voltage gain from base to collector.
Sometimes the emitter resistor has a capacitor across it, so that the AC gain can be higher than the DC gain.