Looking at the video you provided, I would tend to think it's something along those lines:
- A particle emitter, to which all threads are attached, is moving around. I haven't really figured what its motion is dictated by.
- Each thread is made of particles flowing away from the emitter. The thread has a direction, used for the initial speed of the particles. Once created, they are independent from the emitter.
- The particles use a quad or a rectangle textured with a fuzzy ellipse shape (you can see them individually when a thread suddenly changes direction) oriented in the direction of the motion. Their color is based on the duration since their creation, and they use additive alpha blending (the saturation to white is typical). After a certain duration, they fade out and die, only to be recycled into new particles.
- I suspect the whole thing might be in a vector field, or something similar blowing wind onto the particles, which would explain why the threads sometimes suddenly change direction.
- There might be some feedback post-processing effect to smooth everything out.
The Flurry screensaver written by Calum Robinson is available as a part of the XScreenSaver package. You can download its source code from https://www.jwz.org/xscreensaver/download.html
The flurry* files are in the hacks/glx directory. It's not an easy job to reverse engineer the algorithm from the source code, but debugging might help.