The primary function of the Trebuchets is to open the two city gates that block Lara's progress. They can also be used to get rid of the Deathless attacking Lara. They are also the only way to light the Signal Fires on top of some of the towers in the city in order to successfully finish the Burn Baby Burn Challenge.
To operate the Trebuchet Lara has to climb up the round platform on which it is located and stand next to the gears on the left side. There she can use her Axe by pressing X to turn the gears left or right using the left analogue stick. A beam that protrudes from the front of the Trebuchet helps aiming. As soon as the Trebuchet is correctly aligned with the target, press and hold the right trigger. Let it go when the mark is on the target. An amphora of Greek Fire will be slung from the Trebuchet and set everything on fire in a wide radius around the mark.