Learn French: Je suis triste de quitter cette auberge.

Because the person who is sad (Je) is the same person who is leaving. In French, when the subject of the feeling and the subject of the action are the same, you avoid using que + subjunctive. Instead, you use de + the infinitive verb (quitter).

How would this sentence change if I were sad that someone else was leaving?

If the subjects are different, you cannot avoid the subjunctive. You must use que followed by the subjunctive mood. For example, if you were sad that he was leaving, it would become Je suis triste qu'il quitte cette auberge.

Why do we need the word de before quitter?

The adjective triste requires the preposition de when it is followed by an action (a verb in the infinitive). Many adjectives expressing emotion, such as heureux (happy), content (glad), and triste (sad), follow this exact pattern: adjective + de + infinitive.

What exactly is une auberge?

Une auberge translates to an inn or a hostel. You will very often see it in the phrase auberge de jeunesse, which means a youth hostel.