Please explain how the gain of this CE-CC circuit is about 3.3, or R3/R1. No very brief answers please, I want to know how it works so describe in detail how you arrive at your conclusion.

 

The right hand side of R1 is fixed at approximately 0.65V (the base-emitter voltage of Q1) Therefore, if the input on the left hand side of R1 is raised by 1V a current of 1mA (1v/1k) will be introduced through Q1. This 1mA current through Q1 will be "mirrored" by Q2 (this is a current mirror) causing a reduction in the voltage at the collector of Q2 of 1mA x 3k3 = 3.3V. The emitter of Q3 simply follows its base.