A cyclotron is a particle (lets say electrons for now) accelerator.
The DEES are isolated from each other and an AC voltage is connected to the DEES.
The DEES are placed between the poles of a (large) magnet. The source of electrons it at the centre of the DEES and, because of the applied voltage, electrons will be drawn into one of the DEES. Within the DEES there is no electric force on the electrons, they are deflected into a circular path by the applied magnetic field. Effectively they travel in a semicircle and come back to the gap between the DEES. The AC voltage will have changed by a cycle and the electrons will be accelerated across the gap to enter the other DEE...travel in a semicircle to come back to the gap (with a larger radius because they are now faster)...the AC has changed...the process repeats.
Acceleration of electrons only occurs when they arrive at the gap between the DEES.
within the DEES electrons travel in semicircles, because of the magnetic field, of increasing radius.The electrons can pass around the DEES thousands of times (AC frequency) experiencing acceleration each rotation.
very much simplified description with no maths. If you are able to cope with the maths look into it !