From the same article you linked, it states
but all it does is place a star next to them in the menu so that you can find them more easily.
With this, you can sort your items as “Favorite First” so that these items will much easier to find.
I personally use this to have my breeding items, such as Everstone and Destiny Knot, to always appear first in the list to find them easier when switching out Pokémon in the day care.
The only snag to this though is that if all of a single favorited item are removed from your inventory (like giving all Everstones to Pokémon or feeding all berries), you have to re-favorite the item. So it’s best to keep at least one of your favorited items in your inventory