It looks like they've made a simple lyric video and a video with different flashing colors and effects and clouds. There are heaps of "backgound" effects like this (clouds, oceans, patterns) available online specifically made for people to use in videos like this. Some are free to use and some like you to pay a fee. You'll even find heaps on youtube or you can spend an afternoon walking around town and filming your own (buildings, traffic, the sky).

Anyway I would say they have then played both the lyric video and the background videos back on a video monitor or TV and filmed that TV playback with a handheld camera (deliberately wobbling and moving the camera about). I'm not sure if there's a specific editing effect that does this for you but it's easy to do manually yourself.

They've then taken the original videos and the videos shot with the camera and just crossfaded them all together either at random and/or at whatever combination they thought looked good. One of the videos has had a round "iris" border put over it (most video editing software will have something similar available).

It's pretty easy to do.

By the way if you plug a video camera straight into a TV/monitor and get it to film what it's recording/playing back at the same time you get the "howlaround" effects that they used to create the early Doctor Who opening titles. It's good fun to play around with and of course with even a basic video editing suite you can enhance the results with all sorts of colorful effects.