The main reason for Shiny Pokemon is mainly to make the game more appealing, and in a way, more interesting. This encourages you to continue playing the game, and to do so they have a low chance of appearing (however, if you are not happy with the 1/8192 chance, use Masuda Method to make it 4/8192). It could also be looked at as an acomplishment - since Shiny Pokemon are so rare, you would certainly feel good about yourself finding one (not to mention bragging rights!). It also keeps the game unique, giving it a vital selling point.
There isn't any correct reason for why Shiny Pokemon are included in the game, though it can be looked at as a way to make the game last longer in terms of '100% completing the game'.