When the light bounces off of the target surface it scatters. The light which bounces back to the sensor array has time offsets depending on the geometry of the scene, those can be modeled as curves.

Similar (in that it's the inverse case) to a bucket of water, if you drop something in it, it creates rippled waves. If you only observe the ripples in the bucket, you can reasonably derive where the thing was likely dropped.

This site — which appears to be the source data for OP as well — has some really nice visualizations. It looks like it's more a scan than a flash, combined with extraordinarily fast frame rates to precisely measure the light's travel time and allow for reconstruction.

http://www.computationalimaging.org/publications/nlos-fk/

Imagine you have a cave the ceiling of which is covered by a thin film of water. At t=0, there is an explosion which shakes the water and make it fall vertically to the ground as droplets.

You have a grid of water detectors on the grounds, which can tell you where and when a droplet hit the floor.

If at t1 you detect a drop of water at floor coordinates (x,y), you know the height of the ceiling at (x,y) is h1 which you can compute from t1. So you can make an image of the ceiling.

The paper algorithm is a fancier version of this, where the droplets are not falling straight down to the floor, but spread out like sphere ripples. So you need some fft math to deconvolve.

The added benefit of using waves, is that you can see behind the objects, and you have a real 3D and not a 2.5D image.

My guess: echolocation, but with electromagnetic waves (light) rather than sound waves

Imagine turning a wall into a mirror. Essentially, it uses a laser beam and fancy math to see around a corner, similar to looking at a mirror.