First, for the wires; the heavy black and red wires go to the battery. They can stay connected all the time (but a disconnect switch is recommended). The small blue wire bunched with these can go to a small switch, then to the red battery wire. When the switch is on, the controller will be powered. Some people just tie this permanently to the red wire and use a large switch to the battery that can handle the full power.
The other blue wire could be several things, but to tell, you should connect a battery or power supply and turn the controller on. Then measure the voltage between battery negative and the remaining blue wire. This voltage will give us a clue. My guess is it will have 5-12v, being a brake wire that gets grounded. If it has battery voltage, then it could be used to drive a battery meter. If it has zero volts, it could be a 'high level' brake line that gets activated by the power from a brake light (usually battery voltage).
A resistor throttle (or pot throttle) works like a voltage divider. You put 5v on one side and the signal from the wiper will go from zero to 5v as you turn it.
A hall effect throttle uses a linear hall sensor (looks like a little transistor) and magnets attached to the moving part of the grip. As the grip is twisted, the magnetic field at the sensor changes and varies the output. Hall throttles have some theoretical advantages. One, there are no wearing parts other than the grip itself, so it should be good practically forever. A resistor throttle will eventually wear out. A hall throttle could be made waterproof (but most are not). In theory should work fine underwater.
The output of a hall throttle is not zero to 5v. Most have a nominal range of 1v to 4v. The controller is designed to take this range. You could use a resistor throttle, but as you twisted it, nothing would happen until you went pretty far, then it would go from zero to full speed over a short range. This could be fixed with a couple resistors if you wanted to use one.