Harry Potter first encounters the Thestrals when they're pulling the carriage up to Hogwarts castle. He wonders what the heck this creepy black winged horse is doing at the front of the carriage, but he's in for a bit of a surprise:
HARRY POTTER: What is it?
RON WEASLEY: What's what?
HARRY POTTER: That. Pulling the carriage.
HERMIONE GRANGER: Nothing's pulling the carriage, Harry. It's pulling itself like always.
LUNA LOVEGOOD: You're not going mad. I see them too.
Yup. They're called Thestrals and they're only visible to people who have seen someone die. Harry just watched Cedric die a few months ago so it follows that he can now see the horses. Luna saw her mother die when she was nine, so she's in the club, too.
Thestrals are obviously a symbol of death, but that doesn't mean they're ominous or scary. Luna tells Harry that they're "quite gentle, really, but people avoid them." Yeah, people tend to shy away from things that remind them of their own mortality.
Seeing the Thestrals is yet another thing that sets Harry apart from his peers. This isn't just another fun year at Hogwarts where Harry's gonna be playing Quidditch and turning mice into tea cups. He saw one of his classmates die last year. Voldemort is back. This stuff just got real, son.