hi b,
Consider the case where that circuit had a single positive supply.
In order to reproduce a close to ground low level input sine wave signal to the base, at the emitter, the designer would have to consider the effect of the required Vbe voltage.
For the transistor to be biased ON, Vbe would have to be approx +0.65V, so this would mean the input signal would have to be superimposed upon this Vbe voltage.
To do this the designer would have to AC couple the low level signal and apply a bias current into the base of the transistor, to bring it into is operating/linear conduction.
By adding this bias and Vbe the transistor will be conducting and there will be a voltage drop across the emitter resistor.
Assuming the bias is correctly set this would mean the output sine wave is superimposed upon the emitter voltage.
By using dual supplies the bias and operating point of the transistor can be chosen to avoid the need for AC coupling and the biasing.
Do you follow that OK.?
EDIT:
Added images.