The Shiny Charm increases the chance of encountering a Shiny Pokémon, both in the wild and through breeding. This is done by generating two extra personality values to attempt to find one that is Shiny, effectively resulting in a tripled chance to find Shiny Pokémon compared to the base chance; from 1/8192 to approximately 3/8192. This stacks with most other modifiers which increase the Shiny encounter rate, such as the Masuda method.
Generation VI onward
The Shiny Charm increases the chance of encountering a Shiny Pokémon from 1/4096 to approximately 3/4096.