All electric keyboards exploit the simple principle of shared frequencies. Just as sounds have specific frequencies, electricity has specific frequencies as well--how many times per second it moves through the circuit. Electric keyboards take the frequency of the electricity, as well as the strength of the current, and translate it into sound. This is how the electric keyboard is able to imitate so many different instruments: careful research has been done to measure the exact frequencies generated by those instruments.
All electric signals from a keyboard start with the oscillator. More expensive keyboards will have several different oscillators, all set to different frequency ranges. When a type of sound is selected, the oscillator will begin generating a specific range of frequencies. If you listened to this sound right at the oscillator, it would sound like a pure tone: all frequencies and harmonics are present. The oscillator is more than capable of generating several tones at once, making tri-tone and other musical effects possible.
If you listen to just the oscillator, it sounds like a smooth tone; all frequencies and harmonics are present. In order to make the note sound like a specific instrument, the keyboard will use filters to remove or add certain frequencies and harmonics. A signal can go through several filters, some of which can be chosen or controlled by the musician, before being passed on. Some filters may also add tones that the oscillator cannot replicate.
At this point, all of the work done to this signal is actually very faint. There isn't a lot of power behind it. This is the job of the amplifier. The amplifier takes its name from "amplitude," the height and depth of a wave. The oscillations of the electricity have created a mathematical wave, and the amplifier will increase the amplitude of that wave. Amplitude in sound notes power, and the higher the power, the higher the volume.
When the speakers receive the signal, they begin vibrating according to the voltage, pushing air in and out. This pushes energy into the air and creates sound. If you looked at the signal being sent to the speakers as a graph, you would see a continuous wave. If you measured the power of the sound reached your ears as a graph, it would match the signal. This is how a keyboard can create such amazing sounds.