Electric guitars are a great example of an instrument that has a lot of physics going on behind it, from standing waves to electromagnetic induction. The latter is especially important in understanding how guitar pickups work.
Guitar pickups come in many different shapes and sizes, but they all work because of one fundamental principle: Faraday's Law of Electromagnetic Induction, a law the states that the EMF induced in a circuit is proportional to the rate of change at which magnetic field lines are cut. Once you understand this concept, the rest is fairly simple.
Guitar pickups are nothing but small electromagnets. They usually use small permanent magnets like alnico (an alloy of aluminium, nickel and cobalt) with fine, thin wires that are wrapped around these magnets thousands of times. The pickup creates a magnetic field which magnetizes any material that is within the proximity of its magnetic field, and in the case of electric guitars, that material is guitar strings. The guitar string is temporarily magnetized due to the pickup. When struck, the guitar string oscillates, and the string's magnetic field induced by the guitar pickup oscillates along with it. This is where we apply Faraday's Law that we had mentioned earlier. The thousands of loops of the coil around the pickup are cut by the moving magnetic field of the guitar string, and hence, there is an EMF induced in the pickup, which is synonymous with the output by the pickup. This output can be increased in several ways. Increasing the strength of the magnet would create more magnetic field lines, and hence more field lines would be cut per second, and there would be more EMF induced and higher output. The number of loops around the pickup would also increase the output by the same logic. Faraday's Law also explains why when the guitar string is struck softly, no sound is heard from the guitar amplifier. When struck softly, the magnetic field of the guitar string may move too little, and loops around the pickup may not be cutting the magnetic field lines of the guitar string at all, resulting in 0 output from the guitar pickup.