In this episode, one character is only able to communicate through singing. That is, she sings what she wants to say and can only understand what is sung back to her. In one scene, Dr. Magnus is searching through her dead father's old scientific records, looking for a clue to a cure. She feels her father's presence as she works, so he seems to sit beside her and sings this song.
By the way, the title "Fugue" is appropriate in two ways. First, a fugue is a musical form in which different voices develop the theme. That, more or less, describes the episode. Second, a fugue is "a rare psychiatric disorder characterized by reversible amnesia for personal identity." That describes the main character of the episode. Well done, writers!
I suspect that this episode was written so that the actors could display their vocal talents, and provide a bit of a break from the show's usual structure. I think it may have been inspired by the incredible all-singing, all-dancing episode of "Buffy the Vampire Slayer" called "Once More with Feeling" (Season 6, Episode 7, 2001). Here's a sample of my favourite cut, "Sweet's Song," unfortunately without the tap dancing visuals.
I don't know if all long-running series feel the need to break out of character or break out of the norm, but it seems Star Trek did. How else would you explain the "Mirror Universe" episodes, beginning with "Mirror, Mirror," in which all the characters get to play their evil twins. Star Trek: Enterprise gleefully adopted the same relief from unrelenting niceness, as the opening for the episode "In a Mirror, Darkly" makes clear.