Spoilers through Harry Potter and the Order of the Phoenix.
Harry Potter (Daniel Radcliffe) can see Thestrals because he watched Cedric Diggory (Robert Pattinson) die. Luna Lovegood (Evanna Lynch) tells him why: “They can only be seen by people who’ve seen death.” The creatures arrive with the loss Harry is still reliving.
Harry sees something his friends cannot
Harry Potter and the Order of the Phoenix
Harry asks what is pulling the carriage and gets a puzzled answer: “Nothing’s pulling the carriage, Harry. It’s pulling itself, like always.” He has come back from a summer of nightmares, a Dementor attack and a hearing at the Ministry, and now he is seeing things nobody else can.
Luna can see it too.
“You’re not going mad,” she tells him. “I can see them too. You’re just as sane as I am.” After months of people refusing to believe what happened to Cedric, Harry has someone looking in the same direction.
Luna talks about death as if it were ordinary
Harry Potter and the Order of the Phoenix
Luna knows what they are and speaks kindly of them: “They’re quite gentle, really, but people avoid them because they’re a bit... different.” It could be a description of Luna, whom other students call Loony.
Her mother “did like to experiment”, and “one day, one of her spells went badly wrong. I was nine.” Luna is matter-of-fact about it: “I do feel very sad about it sometimes, but I’ve got Dad.”
She gives Harry a way to talk about someone who is gone, beside a creature that looks far more sinister than it behaves.
Harry was too young to understand his first death
J.K. Rowling, “Thestrals”
Harry was a baby when he saw his mother killed, so Cedric is not the first death he has witnessed. In “Thestrals”, her official essay on the creatures, J.K. Rowling explains that seeing a death is not enough. You also need an “emotional understanding of what death means”, and Harry “was barely out of babyhood when the murder happened”.
Understanding that someone is gone for good is the real test. It fits Harry’s infancy and his grief for Cedric, though Luna only gives him the short version.
Harry has already watched Quirrell die
Harry Potter and the Sorcerer’s Stone
Harry faces Quirrell in the chamber beneath Hogwarts. Warner Bros. Pictures
Professor Quirrell (Ian Hart) is the harder problem. At the end of his first year, Harry watches him turn to ash, before the spirit leaving Quirrell’s body rushes through Harry and knocks him out.
Cedric’s death gives the creatures their meaning, because Harry is still reliving it. Quirrell leaves a gap: nobody explains why watching him die does not let Harry see Thestrals until his fifth year.
Harry sees Thestrals because he has seen death and finally understands it, and Luna turns that frightening sight into something two people can share. Quirrell’s death remains unexplained on screen.