To begin with the earth is considered to be a black body. A black body is any material with emissivity= absorptivity. Now, if the earth is seen as a black body then every radiation it receives is what it gives away but when the earth receives shortwave radiation(SW) from the sun it does not emit everything. The earth absorbs the SW radiation and emits longwave radiation. This radiation doesn't also totally leave the Earth's atmosphere but it is blocked by clouds which re-emit it to the earth. This continuous process is the energy balance.
The energy balance is significant because it drives weather patterns, it causes pressure changes, influences evaporation and condensation rates and determines the depth of the atmospheric boundary layer. It is necessary to know the energy balance as excess energy or insufficient energy will be hazardous to the earth. A classical example of the effects of this is climate change.