Borderlands 3 is filled with stats, but most of them will be obvious and self-explanatory. The Luck stat can cause a little confusion, as it is not expressly said in the game what this stat will have an effect on.
The Luck stat will dictate the level that your gear will drop at. Every time you kill an enemy, a loot table is rolled. Depending on what you killed, where you are, and what difficulty level you are playing on, these loot tables can drop different items, from Common to Legendary. The higher your Luck stat, the better the chance you have of getting better gear.
There are two ways to increase your Luck stat. The first is by finding loot that gives you some kind of bonus to your luck. The second is by earning Guardian Points. When you finish the main story of the game, you will open up Guardian Ranks. Earning experience will increase your Guardian Rank, earning your Guardian Points that you can spend on specific trees, auto-generating a selection of buffs that you will need to pick from. Some of these can be increased Luck.
By stacking various sources of increased Luck you can give yourself a nice boost when it comes to getting Legendary weapons, but the exact math behind it all is currently unknown. It is just a case of the higher you can make the stat, the better.
Some items that give you increased Luck may need you to sacrifice another stat, so you should always carefully consider the benefits of this type of trade-off before taking on some of the game’s tougher challenges.