Why does adding a capacitor increase voltage gain?
With a transistor having reasonable current gain, configured as a common emitter amplifier, the AC gain can be approximated by the ratio of the collector resistor to the emitter resistor. The capacitor across the emitter resistor greatly reduces the effective emitter resistance at frequencies where the capacitive reactance is less than the emitter resistor. Note that the input signal, Vin, basically appears across this emitter resistor, Re, so the emitter current is given by Vin/Re.  This current, again for a reasonably high transistor current gain, is the same as the collector current.  Thus the output signal is this current multiplied by the collector resistor, Rc, which is Vin(Rc/Re).  Hence, as already mentioned, the voltage gain is Rc/Re. The capacitor reduces the effective value of Re, hence increasing the AC gain.