Automatic gain controller gain action
Can you guys explain to me how the turning on of Q2 and the charging
of C2 increases the vgs of the FET?

The text in your question didn't say that \$V_{GS}\$ increases - it says that \$\color{blue}{\text{the gate voltage increases}}\$ and this is an important distinction because, as the gate voltage increases from a negative voltage towards 0 volts (the source voltage), \$V_{GS}\$ is actually decreasing in magnitude.

The JFET gate is naturally biased at -Vee when there is no effect from Q2 i.e. when Q2 is not activating. This natural negative bias ensures that the JFET is fairly deactivated and acts like an open circuit. This does not attenuate the input signal.
As the input signal rises the output AC voltage rises and it reaches a point where Q2 starts to activate. That causes Q2 to conduct and raises the JFET gate voltage from -Vee towards 0 volts. This allows the JFET to turn on more and it starts to attenuate the input signal going into the amplifier.
In effect, it's a crude gain control.