how is this circuit working as an amplifier
Obviously, the transistors are not there to amplify the video signal. Instead, they seem to be functioning as a clamp that keeps the video signals grounded (through two diodes) whenever the red line at the top has less than about 7.7 V on it.

When the voltage is high enough to turn Q202 on, Q203 is turned off and the diode cathodes are pulled up, effectively switching them off and allowing the video signals to operate over their full range.