OK then, if you only want the one extension this will be straightforward. Don't worry too much if you can't read what the cable says on it, I can see that it is twisted pair from the photos so is likely good enough even if it is not CAT5e (CAT5E is significantly better than what is required, the minimum being CW1308 telephone cable).
First, make sure you know which pair of wires are the ones going to the A & B terminals on the back of the master socket. Be very sure not to accidentally cut these as they are not your property, they belong to Openreach. You should be able to gently pull all the other wiring forward slightly and untwist things in order to improve access. You will note that each grey cable has a solid blue wire, a solid orange wire, and a blue with white stripe wire (plus a bunch of other wires which have been cut off flush with the cable end). Remove the existing three wires from the three cam-lock* terminals on the front of the master socket.
The task now is to identify which of the five outgoing cables goes to the extension socket you wish to use. Then connect the solid blue wire from that cable (only) to terminal 2 on the front of the master socket and the blue with white stripe wire (from the same cable) to terminal 5 on the front of the master socket. The solid orange wire and corresponding terminal 3 are for the ring wire and should be left disconnected.
There's a few ways of identifying which cable is which. The simplest would be by trial and error, so just cut all the wires out of the crimp terminals, wire up a single pair to the front of the master socket, replace the faceplate, then see if a telephone works at the extension socket you want (could also try all the other extensions then label all the wires in case you change your mind in years to come). If you don't have a telephone this last step could be done by checking for around 50V DC across terminals 2 & 5 behind the extension socket, but don't use your router to test it at this stage as repeated connections and disconnections can lead to Openreach's equipment thinking your line has a fault and banding your connection as a result.
Another technique you could use, if you would prefer not to separate out all the wires, would be to short the spare green and green with stripe (say) wires together at the extension socket, then check for continuity between these spare wires behind the master socket using your multimeter. Once you have located it, then simply cut out the relevant blue, blue with stripe and orange wires from the crimp terminals and wire up the blue ones as above.
When wiring up the terminals, try to keep the blue and blue with stripe wires twisted together (in the same way the spare pairs behind the extension socket are) over as much of their length as possible.
Once this is done, tidy up the cut wires somehow towards the back of the box, replace the master socket, replace the faceplate and connect your router to the extension (no filter required here if you don't use a telephone at the extension). Please let us know the result.
* These cam-lock connectors work by hinging them up from the bottom. Once it is lifted, you can pull the wires out. To install new wiring, insert straight pieces of wire into the relevant holes (without stripping any insulation) then fold the connector back down. The connector contains metal teeth which will cut a hole in the insulation as you fold it down in order to make contact.