The way out is on the high ledge at the far end of the room. To open it you must activate the 3 square pressure pads on the long ledge near the entrance. The left and right pads control the huge gears on each side wall. The center pad actually raises the door. All three pads must be depressed at the same time for the mechanism to work. You'll need 2 of the 3 movable cages. Lara will press the third pad.
This sequence is also shown in a gameplay video.