Because it's your job description? You're told outright the moment you sign up at the Adventurer's Guild right at the very start of the game that being an adventurer isn't all fame and glory and saving the world, it usually just means doing mundane, boring tasks for the average person in the street for little reward, but by doing such odd jobs you build a rapport with the people of Eorzea, and eventually you're entrusted with more important tasks.

Also there is a technical reason why there is so many filler quests - the main scenario was designed to be the main form of levelling originally so the story had to be executed in such a way that there was enough quests to get a new person to 50 by the Ultima Weapon, so some padding was required. Of course things have since changed with alternative means of levelling (such as dungeons having their exp boosted, and of course Palace of the Dead), but the main scenario is still considered the primary means of levelling by the development team, and this will not change.