These other answers are, I think, missing the main boon of Extreme Evoboost. Now let me preface this that this is a gimmick strategy that is shut down by Prankster Taunt, good leads or even something fast enough to KO Eevee, making it difficult to pull off. However, it is possible.
This strategy is for Doubles or VGC. You'll want to lead with Eevee and a Pokémon with either redirection or Fake Out. The idea is to stall the enemy until your Eevee can get off an Evoboost and then Baton Pass it to something scary.
Redirection moves are those that force attacks to target a certain Pokémon. Follow Me, Rage Powder and Spotlight are all redirection moves. The reason you want these is to make sure that your opponent doesn't KO Eevee before it can pass along its boosts. You'll also want to make sure your redirector is bulky enough to survive attacks aimed at Eevee. Here are some option for redirection:
- Clefable gets Follow Me and resists Fighting moves that will target Eevee. It also gets Spotlight, a new redirection move that forces moves to target any Pokémon on the field, including your opponents. Clefable is also really bulky and just a good Pokémon overall. Celesteela is a huge problem for Clefable though, so you should pray that Spotlight keeps Heavy Slam at bay.
- Fake Out causes one target to flinch and is a decent alternative to Spotlight/Follow Me if you can't fit Clefable on your team. Pokémon that can learn Fake Out include Raichu, Hariyama, Sableye, Kangaskhan, and Weavile.
Finally, you're gonna want to baton pass onto something that becomes extremely scary with +2 in every stat. Okay, pretty much anything becomes scary then, but here are some prime candidates:
- Tapu Lele
- Tapu Koko
The idea of the Extreme Evoboost strategy is to get off a Evoboost + Baton Pass, then proceed to bash in faces with your rampaging nanoboosted sweeper of choice. If you have any further questions, let me know!