You kill a monster. Items drop out of said monster. Between those two actions, a chance roll occurs. The item that drops, let's say it's a sword, has a chance of being either Normal, Enhanced, Rare, Legendary, or Unique. Now for a better way to represent this:
PLEASE NOTE THE NUMBERS USED HERE DO NOT REPRESENT ACTUAL CHANCE RATES.
THEY ARE ONLY BEING USED TO ILLUSTRATE HOW LUCK WORKS.
180 - Normal
14 - Enhanced
3 - Rare
2 - Legendary
1 - Unique
180+14+3+2+1 = 200
Let's say these are the numbers for when you are wearing no Luck items. They do not represent the game's drop rates and are only used for this explanation. This would mean that the sword which is about to drop has a 90% chance of being Normal, 7% Enhanced, 1.5% Rare, 1% Legendary, and 0.5% Unique.
Now, let's say you're wearing Luck gear totaling 80 Luck. This subtracts 80 from the Normal number which is now 100 in the case of my example numbers. Suddenly:
100 - Normal 14 - Enhanced 3 - Rare 2 - Legendary 1 - Unique
100+14+3+2+1 = 120
This means that now there's only an 83.33% chance of a Normal roll while all of the higher rarities increase in their percentage chance: 11.67% Enhanced, 2.5% Rare, 1.67% Legendary, and 0.833% Unique. Sounds pretty good, right? Hold tight, here's where it gets interesting.
Monsters which are Rare, Legendary, or Bosses are already set to drop selections from different rarity groups. To explain, let's say you're about to land the killing blow to Shulgoth. This time there's a series of chance rolls. This is because monsters who have a rarity level higher than Normal have multiple chance rolls when it comes to dropping loot. For example:
Chance Roll #1: Normal -> Unique 2 items from this group
Chance Roll #2: Enhanced -> Unique 1 item from this group
Chance Roll #3: Rare - > Unique 1 item from this group
There are a lot of percentage chances at play in these rolls, naturally. However, Luck only affects Chance Roll #1. Why? Because it's the only one that has Normal items as a possibility.
SO! Luck improves the quality of your drops by reducing the percentage chance that a drop is Normal quality. It does not directly improve the percentage chance that a drop is higher quality but instead indirectly. It also does not increase the number of drops in any way. Nor does it improve chance rolls where Normal items are not a possible drop.