In the simplest terms, the Schumann Resonance is the resonance of low frequency electromagnetic waves bouncing around in the earth, with the earth being the active "resonant chamber."The earth is surrounded by a layer of charged particles called the ionosphere. This layer of active charged particles is able to reflect low frequency electromagnetic radiation in a similar way as that of a mirror which is able to reflect light. Electromagnetic radiation is like radio waves or light. It travels at close to the speed of light, 300,000,000 m/s.
The ionosphere's property of reflecting radio waves is what enables you to tune in to radio stations around the world. If the ionosphere did not reflect radio waves as it doesn't reflect light, we would not be able to tune in a distant radio station because the signal would have gone out into space. This is exactly what happens with high frequency electromagnetic radiation like light. It is what allows us to see stars "through" the ionosphere.
The earth's surface is also conductive and, therefore, also reflects low frequency radiation. It is in this way that radio station signals from around the world reflect back and forth between the ionosphere and the surface.
So, both the earth's surface and the ionosphere are conductive surfaces able to reflect low frequency electromagnetic radiation. Together they create a resonant cavity. This cavity then is what makes the earth into a "resonant chamber" for electromagnetic radiation.
The earth is approximately 6,371 km in diameter or 40,030 km in circumference. An electromagnetic or radio signal travels at close to the speed of light. It, therefore, takes close to 0.1334 seconds to travel around the earth. This time would correlate with about 7.49 Hz. Notice that this is less than the 7.83 Hz that is commonly quoted. If you take into account that the ionosphere is approximately 300 km up and adjust the radius by 150 km, the calculation results in a round trip taking 0.136 seconds, which correlates to a frequency of approximately 7.32 Hz.
The 7.83 Hz is not an exact, specific frequency, but the "average" peak frequency of the resonance. It is necessary to average the Schumann frequencies over long periods of time, since the resonant peak is not particularly distinct and it changes with environmental conditions. This is the reason there are many stations like the one at Stanford that measures the frequency daily.
It is also interesting to note that 7.83 isn't the only resonant frequency. Other "harmonic" frequencies are also resonant, including approximately 14, 20, 26, 32, 37, and 42 Hz etc.