The red and black wires that go to the yellow on the throttle are battery voltage for the battery voltage indicator LEDs on the throttle.
The pedal sensor takes its 5v from the grey wire (might be white), which is spliced off the throttle's 5v for some reason, which is unusual. I think it's done like that so that you can use the throttle switch to switch off power to both the throttle and pedal assist. The switch on the throttle is most likely on the throttle's 5v line. The splice is normally done inside the controller on other controllers, so you get three wires to the pedal sensor.
Apart from that, it's standard ebike wiring. It's a bit crude by modern standards, in that there is only one level of power with the pedal sensor, and that's full power. You can improve it a lot by using a modern controller, like a KT sine wave one from PSWPower plus LCD with 5 levels of assist. It will also go a lot better if you upgrade to 36V , which will give 50% more power and 50% more speed.
The standard controller will work at 36V, but the LEDs on the throttle won't show correctly, as they will always be on, and the controller won't cut off when the battery goes too far down, so you have to be careful about that. If the present battery is U/S, it's best to replace with a 36V one and upgrade the control system at the same time - total cost about £250.
