You can challenge this dungeon twice a day to earn rewards. To unlock new stages, you have to clear a stage with 3 stars. Each stage has its own condition to get 3-stars. The rewards include Sanctuary-exclusive weapons, equipment, and materials. This is a 3v3 battle. You can tackle this with other players or attempt to go solo with AI-controlled allies, which you can select from your bonded NPCs.
The Sanctuary gears you obtained can be equipped from the Sanctuary Gear tab in your Gear Menu. While these gears are only exclusive when doing the Sanctuary Trials, they can still provide a huge CP boost so doing these trials is necessary to keep your CP up. Both your Main Class and Secondary Class will have its own Sanctuary Gear slots. Once you have your sanctuary gears, you can further enhance them using Sanctuary Forger Stones, which you can also get from Sanctuary Trials.
During battle, your combat mode is set to Manual. However, you can still toggle it to Auto, should you wish to. If you have AI-controlled allies, they’ll move and attack on their own so you just need to worry about your character.