swalve is correct, the camera takes about an hour to capture an entire scene. To quote the video (starting at 1:32):
"So what's happening is, the camera keeps taking images and we very slowly rotate this mirror to scan our field of view across the entire scene. And because all of our pulses look the same we can in the end go and combine all these images we took to get one complete movie of the scene."
That movie of the light pulse had a different, but identical, pulse in each frame.
So what they have invented is a very accurate shutter trigger and a sensor with a very fast shutter speed.
The image sweep is accomplished by mechanically moving a mirror very accurately, but not quickly.
This camera rig is not really a slow motion camera, it does not capture a single event at 10^9 frames per second.