Bond quests are expeditions into stories outside the tower where you meet named mercenaries. Your highest cleared floor unlocks quests, and completing both combat chapters lets you receive that person as a recruit candidate.
Open "Bond Quests" in "Dungeon" to review available stories and their recommended difficulty. Prepare a party for the quest, then use the recruit-candidate claim after completing the story.
Take a quest to seek a new member for your party or learn that person’s story. Sign the rescued candidate in "Recruit"; if the special-candidate slots are full, resolve a waiting offer before claiming them.