Symbolism, Imagery, Allegory
Upon Dumbledore's death, Fawkes the phoenix sings and then flies away for good. As you know, a phoenix is a symbol of rebirth – it dies by bursting into flames, but is then reborn from the ashes. Dumbledore was all about giving people second chances – he was ready to give Voldemort a second chance when he returned to Hogwarts ten years after graduating, asking for a teaching job. And he certainly put his trust in Snape when others would not. The Phoenix is a powerful force in the world of Harry Potter for its reminder of the circle of life. Death is a necessary part of life. Even magic cannot combat death. This information stands in strong relief to Voldemort's efforts to conquer death.
Still, we are filled with a sudden sadness and emptiness when Fawkes leaves Hogwarts for good after appropriately mourning Dumbledore's death. The silence that follows is a heavy one, and we are left to wonder what comes next once Dumbledore is gone. Who else is there who believes so fully in the good of people, who believes in giving people a second chance? There's a finality to Fawkes's departure that suggests something more than Dumbledore's life has come to an end.
You might want to think about where else we see the phoenix in the Harry Potter series. What other symbolic weight might Fawkes have?