I'd guess the connections from the dartboard are just an array or grid of switches.
The two connectors they go to appear to partly also connect to the extra buttons on the circuit board, so it's likely both the sensor and setup contacts are all read in the same way.
Look at how to read a keyboard matrix - eg. articles such as this:
A Keyboard Matrix adds a bunch of buttons to your project without burning up all of your I/O pins. Here's how to build a pushbutton matrix for an Arduino.
The harder part will be finding out which combination of connections each dartboard segment switch connects to...
I'd keep the section of the circuit board that has the sockets for the flexible cables and remove the area with the controller IC (the black blob).
(Don't try to connect to the flex cables other than by using the connectors on the PCB; the cable is rather fragile).
Fix the pcb somehow so the flex cables are not getting moved about too much or strained.
Then solder wires to all the connector pins on the underside of the board and label them all. Draw out a 10x10? grid on paper
It looks rather like one cable & connector are the matrix rows and the other connector matrix columns, so each switch should connect a wire on one connector to a wire on the other. Label the wires and paper grid rows/columns to match, eg. letters on rows and numbers on columns.
With each dartboard segment pressed in one at a time, go through the combinations or pins/wires to "map out" the switch to connector arrangement.