Outbreaks are in-game events in which a large number of wild Pokémon of a single species spawn in a randomly selected area. These events significantly increase the chances of encountering that species, making it easier for players to catch, battle, or shiny hunt specific Pokémon. Compared to normal gameplay, more Pokémon appear during an outbreak.
What is the purpose?
Outbreaks are especially useful for shiny hunting. The increased number of encounters raises the chances of finding a shiny Pokémon. As players catch more Pokémon within an outbreak, a "shiny chain" can build up, further improving shiny odds.
Will I get a shiny from this?
In very rare occasions, a shiny outbreak can happen!
Do I need to do them, or can I ignore them?
You can simply ignore them if you don't care about the outbreak of that specific Pokémon. Some outbreaks might contain aggressive Pokémon, so it's better to watch out!