This post is about electrical system basics, what powers what and why. Not all aircraft have electrical systems, but the ones that do, it is all the same.
In a car, there is a battery, mostly used to start the motor. When the motor runs, there is some excess horsepower used to drive an alternator that is used to recharge the battery and run all the electrical systems in the car. If it didn't have enough power to run everything and recharge the battery, the battery would always be dead.
In an aircraft there are batteries that will start the aircraft. Your normal GA aircraft will probably have a single battery that will be enough to start the motor. Some will be 12V and others 24V. The voltage doesn't matter the concept is the same. The battery starts the motor, and the alternator powers all the other systems while recharging the battery. Light jets are similar, but some will have a starter/generator. The starter once the engine is going will become the generator, to power the electrical systems.
In larger jets, like the 737, there will be a battery, but it will be used to start the APU. The APU is a small engine that turns a generator. The APU will generate enough electricity to start the bigger jet engines. The jet engines have starter/generators on them, that will be used to power the whole aircraft and recharge the batteries.
Many times, jets will use a ground power unit (GPU). This is a cart with an engine and generator or interface to mains power that will be used to power the aircraft while on the ground, and can be used to start the aircraft.
Once the aircraft is powered, there are many systems. These systems can include avionics, entertainment, lighting, etc. Each of these systems will have one or more circuits. Each circuit will be protected with a fuse or circuit breaker (or an electronic equivalent). The circuit protection device is there to prevent fires. The size of the circuit protector is related to the size of the wire going to the circuit. In an aircraft smaller wires mean less weight, so using the right size is critical.
As current flows through a wire, there is resistance. Copper has very low resistance, but not zero. That resistance to the current flowing translates into heat. If the wire cannot dissipate the heat generated, it will transfer the heat to the insulation, potentially melting that. If the insulation melts, and maybe melts the wire next to it, or as the wire passes through a bulkhead and can conduct to the metal airframe, even more current will flow causing even more heat, and maybe something near by will catch fire.
The circuit protector should interrupt the current flowing in the wire before the insulation begins to melt. If the current isn't flowing there is no heat. If there is no heat, there is no fire (or melting insulation causing smoke, etc).
Sometimes things go wrong outside of this. UV rays can cause wire insulation to become brittle (older insulation), and then vibrations would cause the insulation to flake off. This may not be too much of a problem, but it may allow wires to touch, and then the current protection may not work properly. Imagine a 20 amp circuit next to a 5 amp circuit, and the 5 and 20 amp wires touch. The 5 amp fuse will do nothing because the 20amp circuit has taken up the load). Similar problems have happened, and fires break out even though the systems seem to be properly designed.
Should pilots be allowed to switch off circuits in flight? If smoke is in the cockpit, and it seems to be electrical, I believe they should be. Should pilots spend any time troubleshooting trying this circuit or that? I don't think so. If a pilot needs something to complete the flight, they should be allowed to try once, but once a circuit is turned off, it should be left off. As much as possible, aircraft systems should be designed to enhance the pilots skills, not override it. When things go wrong, pilots need to land as soon as practicable, and have the systems checked out while safely on the ground.