It's a super annoying minigame, but I finally figured it out. You have to spam the click, and keep clicking even when the circles aren't shown. And when the circles do appear, prioritize the reddest ones.
In any case, the demo is really short. Ya didn't miss much if you stopped after that.
Just a small bug report: on the current public build (0.3.34), Nim's full-size sprites are broken in some of the early dialogues. Specifically, when you go back upstairs to talk to Kate, and if you inspect the cat-girl picture in the naturist's house in Bushwood.