The system can basically be described as a wireless guitar lead. It replaces the job done by your guitar leads, but with less wires. Instead of the physical cable it uses radio waves to transmit the signal from your guitar (more accurately the pickups) to your amplifier. This is done by a transmitter (attached to your guitar) and a receiver (attached to your amp). The transmitter converts the pickup output into a frequency modulation (FM) radio signal which is then transmitted in all directions and picked up by the receiver which would be on top of your amplifier. The signal picked up will then be converted back into the original guitar signal.
So to clarify, I would say the following processes are followed:
1. Sound output from guitar pickups
2. Sound converted to FM signal
3. FM signal transmitted out in all directions
4. Receiver picks up FM signal
5. FM signal converted back to the original guitar signal by the receiver
6. Sound goes into the amplifier as normal
So as you can see the first and last step is the same as using a non wireless lead, but the journey it takes is just a little different.
Obviously all wireless guitar systems are not all the same, for example some transmitters will be more powerful than others: One could be a little 2 inch wire attached via your cable socket, or another could be a transmitter large enough to necessitate being attached to your belt or guitar strap. Another difference is whether you use a VHF or UHF system, or whether your receiver has one or two antennas. These are quite complicated areas which will need their own discussions so ill discuss variations such as the ones mentioned above in another article soon, as I hope to do with digital transmission too. But there you have it; the basics of how a wireless guitar system works; the bridge between guitar and amplifier is radio-waves rather than a guitar lead.