I found many examples where fluorescent lamps begin to emit light near the Tesla coil. As far as I understand, the electromagnetic field affects the gas itself. How can we get this energy for example to power a small fan or other devices?
You might want to look at this article.
The basic idea is that you have a resonant circuit (so the receive circuit is efficient at picking up the incoming EM radiation), and then use a rectifier to charge a capacitor when the voltage in the circuit is large enough. Using a series of diodes and capacitors allows you to "step up" the received voltage to something practical (at the expense of current you can draw).