Physics of every-day life: rotating bag of tea

Of course it has something to do with the liquid water entering the gas phase just above the cup of tea, but how does that give the bag of tea a directed motion to one side?

Nope.

The teabag is dangled by a string. Remember that the string is made of wound up threads:

Now, the threads stay wound up because they fit well and they have a knack of being permanently deformed if held in the same orientation for too long (for example, if you loop a thread and press it for a while, the loop stays). Wetting them undoes this permanent deformation.

When you dip it in tea, however, the liquid is absorbed by the threads, and they expand. There is no longer enough room for them to stay tightly wound. Besides this, they lose the permanent deformation, so they have the motive and the means to unwind -- which is what happens.