How does the diagram really work?
What is the oscillator for in this experiment? Why are two copper
  circles used, instead of directly the resonant coils? Is my
  understanding of the diagram correct?


Remember a setup with two coils is like a transformer with space in between. An inductor with a steady field can't transmit energy to another inductor. To create a voltage change in a loop of  wire the field either
1) Needs to change over time
2) the loop needs to be moving across a magnetic field that has a gradient

In both of these situations the magnetic field is changing, and the voltage across the loop changes with the magnetic field. If the magnetic field does not change, then no voltage change and no energy transfer.


  In addition, there are some symbols that escape me, I think one is the
  resistance and this is to control the voltage, but the other 2 do not
  know what they are, I have been looking for, but no symbols appear
  like those.


The first box means a sine wave generator for creating the oscillations in the first coil. The rest are simply loops.

Look at a Rectifier circuit to understand why you need to convert the voltage back to DC. You don't have to have a rectifier if you need AC current for your load.