According to Serebii:
- Agile Style raises the Pokémon's speed which makes it move sooner but lose power.
- Strong Style lowers your action speed but increases the power and effects such as HP recovery and chance of secondary effect.
Style will usually affect Status moves duration, for example Acid Armor:
Standard: 5 Turns
Agile: 4 Turns
Strong: 7 Turns
In other cases their power, like Roost:
Heals the user by up to 50% of their maximum HP (70% if Strong)
Or even their accuracy, like Poison powder:
Standard: 80
Strong: 100
(Needs more investigation)
Other sources states that status moves are affected in other ways:
Agile status moves have lowered effects.
This tradeoff comes with the added benefit of a quicker Action Speed, which will usually result in allowing you to have two turns in a row.
Strong status moves have increased effects.
This power comes with the trade-off of lowering your Action Speed, which will usually result in allowing the opponent to have two turns before you get to go again.
Some examples:
Strong style: a move such as Howl raises your offensive stats by one stage, it would additionally increase your offensive stats by an extra stage. If you want to use Poison Powder the effect of the poisoning is increased, which means your opponent takes more residual damage from the poison than usual.
Agile Style: the effect of your status move will be decreased: Howl will decrease the amount by which your offensive stats increase, Poison Powder will deal less residual damage than usual, and so on.