The threads disappear, not the thing covered by the threads.
No, the thing covered by the threads disappears also in the video embedded in the source story. Way cool.
Look at the (wooden?) structures holding the threads. They do indeed vanish when the nanotubes are heated.
It looks to me like an old magician trick done with mirrors.
It looks like the cloth is just reflecting the background to the left of the experiment which gives the perception of invisibility. Its all placement and camera angle.