Hyper Cutter prevents other Pokémon from lowering the Attack stat stage of the Pokémon with this Ability. It does not prevent the Pokémon with this Ability from lowering its own Attack, such as by using Superpower.
In Pokémon Emerald, if a Pokémon with Hyper Cutter uses Cut in the overworld, it will remove all tall grass within a two-square radius of the player's position, as opposed to a one-square radius without this Ability.
Generation IV Onward
Hyper Cutter has no effect outside of battle due to the removal of the ability to cut tall grass.