[_Microft]

I think it is either a single layer board or the back side is empty anyways.

The complexity certainly does not require a board with two or more layers. Here is how it is made up:

Positive pole goes to the pins of the connector on the left-hand side, negative pole to the ground fill (look closely to the right connection at the battery and you will see how it connects to the area with the lighter shade of green on the board - this is a copper area shining through the green solder mask).

On the right hand side, the different pins of the connector are wired to the test points, resistors and diodes which at the other end connect to the ground fill. Now if a cable is plugged in both sockets and if there is a connection on a particular pin of that cable, the circuit for the respective LED is closed and it lights up.