As of the Neon Nightsky Patch, this item is now obsolete, as the Loot Collector will do the exact same thing. Any still placed Style Saver will work identically to the "new" Loot Collector, the only difference being the way the Style Saver looks, and the "tooltip" when you look at the Style Saver still says the old "style saving" tooltip even though it doesn't do that anymore.
The Style Saver was an item created at the Crafting Bench that allowed Trovians to "save" the styles of items to use later. Opening the Character Sheet with the C key and clicking on one of the boxes under Weapon, Face, and Hat will allow you to equip any style you have saved.
The Style Saver worked similarly to the transmogrification used in games such as World of Warcraft, but in Trove, a player can equip any known style at any time, it's not bound to a single item.
Using the Style Saver to save unknown styles would give the player 1 Mastery Point.
After the Patch, the Loot Collector got the Style Saver's style saving ability, and Style Savers were "rendered obsolete". All Style Savers in inventories were replaced by Loot Collectors.