For this kind of a shot, it doesn't matter if the flash is first or second curtain because there is no direction to the movement. (ie, the phone could have started on the left or the right side and we'd have no idea of the difference). What does matter is that you have a dark environment so that only the light trail is normally exposing. You then take either a long exposure or a bulb exposure (which keeps the shutter open as long as you hold the button down).

The flash exposes the people and since it is dark, they don't continue to expose much, but the light trail from the bright screen exposes quickly and thus you see the light trail but not much else that was exposed during that part of the shot. It also probably took a fair bit of practice (or luck) to get it to come out that well as typically the light trails would be less even.

It's also possible that they faked the effect in Photoshop since that is one heck of a smooth circle. It's certainly possible to get that effect in camera, but they seem to have done a remarkably good job in many aspects, so either they did a lot of time and patience to get it just right or they blended it in artificially (in part or full).

To produce it in post, using an additive blend would properly mix the light sources in the image with the light source of the rainbow such that the sum of the brightest spots would shine through. Any tool of choice could be used to produce the actual arc.