The lucky number 7.

It works, you just dont understand it.

Every non-boss enemy in the game (Heartless, Nobody, and others) has a set amount of items of which they can drop. Of these are items for Synthesis, and some useful consumable items, such as Elixirs, and Potions.

Lucky Lucky improves the Luck rating, which means that for every Lucky Lucky equipped by the active party, (Sora, Donald, Goofy and/or visiting character), affects your Luck rating, and affects how often said and other items will drop.

In short, more Lucky Lucky abilities equipped means items will drop more frequently.

blah blah blah no one listens to what I say
The math behind Lucky Lucky:

The Strategy Guide errorneously lists Lucky Lucky as improving your Luck rating (the percentage of which items are dropped) as being multipled by 1.3 total. E.g., if a potion has a 10% drop rate, then having one Lucky Lucky will improve it to 13%, then ((1.3*x%) + x%), and so on, and it is your static average (little fluctuation).

This is however, inaccurate, and Lucky Lucky rates are determined by Deminishing Returns, in which if a drop rate begins at 10%, it will then take 10% of the remaining percentage. So if the drop rate is 10%, and you have one Lucky Lucky equipped, the remaining 90% is where you take 10% from, meaning that Luck+one Lucky Lucky will wield a 19% drop as your MAXIMUM drop rate value (LOTS of fluctuation).