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.
Because of the feedback issues that you mentioned in your question, neither in-line amplifiers, nor the transceivers "feeding" them, can operate in a full duplex fashion when transmitting and receiving on the same RF channel. In the transceiver itself, there are duplex switches that switch between the outgoing and incoming amplifier chains that are triggered with a logic signal from the PHY silicon.
Since an external amplifier does not have access to the TX/RX logic signal, an external amplifier has a detector circuit to sense when the transceiver is transmitting, which then provides the logic signal that switches between the incoming pre-amplifier and outgoing power amplifier chains. It should be noted that, along with the quality of the RF design of the amplifier chains themselves, the speed and accuracy of the detector circuit itself can have huge impacts on the performance of the external amplifier.
An (overly)simplified block diagram::
Note: While not shown in the diagram, the RF Detection Circuit controls the two switches for the antenna and the radio, both at the same time.