What does the wifi amplifier exactly amplify?
The question is not exactly about the wifi, but on how does this
  operate in full duplex


It doesn't; read the more detailed specification on the page you linked: -


  Operation Mode: Bi-directional, half-duplex, Auto-Switching via
  carrier sensing.


The full detailed spec is here: -


  Operation Range: 2400-2500 MHz.Operation Mode: Bi-directional,
  half-duplex, Auto-Switching via carrier sensing.Frequency Response: ±
  1dB over operation range.Input Power: 3dBm (Min.)-20dBm (Max.).Input
  Power 5 ~ 20 dBm.Optimal Input Power 9 ~ 15 dBm.Output Power:
  8000mW/39dBm nominal for 802.11b/11Mbps.Connector: SMA Receptacle,
  50ohm.Transmit Gain: 17dBm nominal.Receiver Gain: 11dBm.Receive Noise
  Figure: 3.0dBm nominal.Operating Temperature: -40 to 70
  degree.Operating Humidity: Up to 95% relative humidity.Material: Cast
  Aluminum.


Note that it has a receiver gain of 11 dB (unlike what other answers may say).

On a slightly different subject, I wouldn't buy anything that doesn't have a downloadable data sheet.