It's not the best explanation, but the schematics show all that is needed to make it work.
The 8 wires are connected to 8 outputs of the CD4017. That is triggered from the 555 to give a second or so on each output.
When one wire is high, all the other are low.
On the receiving end any wire that is low is connected via a diode to the common wire in the receiver, to provide an earth path.
A major function of the tester is simply to have the LEDs with numbers against them so that it's easy for the user to work out what is happening.
The tester relies on there being at least one other wire working to give a return path. Also if the cable is wired wrong, with each pin wired to the one above, like this:-
1 -> 2
2 -> 3
3 -> 4
4 -> 5
5 -> 6
6 -> 7
7 -> 8
8 -> 1
that won't be detected by the tester as the lights will still illuminate in the correct order. However, wiring an ethernet cable wrong like that is going to need to be done deliberately.
The tester will be really useful for the usual faults of a wire or two not making connection, a couple of connections swapped, or the far end being from a completely different cable.