This is true! Spiders use their silk to help them "fly" around, mostly you see it with the spiderlings (babies) because they're small enough to ride the wind, but it also happens with very small adult spiders. They lift their bum into the air with a little silk to make a sort of sail and wait for the wind to pick them up. It's called "ballooning".

This spider you've taken photos of is called an orb-weaver and they're harmless, and they are notorious for using this ballooning method when they hatch from the egg sac to find a new home.

It's pretty awesome that there's a song that talks about this spider transport method, I love it when education is transferred in fun forms like songs and games.