When you first prepare to go into a dungeon, you may elect to bring items with you. These will take up your 20+ slots, of course. General items to bring would be food (in case you want healing), and equipment that is better than what the adventurer currently has equipped. The prompt you get at the beginning, if you select yes, auto-equips all pieces of equipment that are better than the adventurer's current equipment. Things like rings will have to be manually equipped though, even if the adventurer has an empty slot.
Adventurers will buy equipment from you, and if it's better they will equip it. But sometimes they buy other stuff, or start selling you things, so you've got a nice shiny equipment that'd look great on them just sitting in your inventory. By bringing it into the dungeon, they can use it.
Keep in mind, though, that such items are thus part of your items carried. If your adventurer falls, you can only take 1-3 items back with you, and such equipment counts. As such, either only bring one piece of good equipment, or have plenty of spares. At the end of the dungeon run, you'll get it back into your inventory as normal for any other item in your bag.