The visualization is known as a chord diagram[1][2], and these particular ones generated by Echo Nest show repetitions in the analyzed songs (e.g. chorus lines). A circle is a good starting visualization because there is a linear progression from beginning to end (single path) and the chord diagram lends itself well to the metrics Echo Nest is demonstrating.
Pretty awesome.. really track dependent though. 'Veridis Quo - Daft Punk', 'Come to Daddy - Aphex Twin' and 'Building Steam With A Grain Of Salt - DJ Shadow' all give pretty great results but something like 'Trigger - In Flames' is.. mediocre at best.
It basically keeps the song going by jumping around to different parts that match within the song. It can lead to interesting effects (different extended solos, jumping around between choruses). It keeps the song going forever.
To start click at the top of the circle after selecting a song.
I've wondered similar but, for Youtube at least, there doesn't seem to be a frame-level API. You can pull a video and analyze it yourself, but you'd basically need to download all of youtube.