The narrator explains that Harry Potter is very unusual: he hates the summer holidays, desperately wants to do his homework, and is a wizard. Late at night, Harry lies in bed under his covers. With a flashlight, he reads through his textbook and tries to write his essay on the pointlessness of fourteenth-century witch hunts. As he works, he listens carefully for the Dursleys. Aunt Petunia, Uncle Vernon, and Dudley are Harry's only relatives, but they're non-magical people (muggles) and hate magic. They'd initially tried to keep Harry's spellbooks from him, but Harry managed to sneak a few out so he could finish his homework and avoid detention.
By introducing the third installment of the series with Harry's homework on witch hunts and burnings, the novel makes it clear that justice will be a primary concern of Harry's third year. Readers even vaguely familiar with the history of witch hunts will know that they were a pointless endeavor that sought to make a show of justice—just as the Ministry of Magic goes on to do throughout the novel.
Unfortunately, Harry is already on thin ice with Uncle Vernon and Aunt Petunia: his best friend Ron tried to telephone early in the summer, but being a wizard, he'd never used a telephone before. Vernon picked up Ron's call and Ron proceeded to yell into the phone and tell Vernon he went to Hogwarts. After that, Vernon yelled at Harry for giving out his phone number and Ron didn't call again. Harry's other friend, Hermione, didn't call either, which has made for a lonely summer. Vernon did decide that Harry could let his owl, Hedwig, out at night, but forbade him from sending letters.
When Uncle Vernon forbids Harry from using Hedwig to send notes, it shows that he understands how important Harry's friendships are to him. This introduces the idea that loneliness is a dangerous state to be in, as at this point, it means that Harry is entirely isolated from his wizarding community and has no idea what's going on in his world.
When Harry gets to a stopping point in his essay, he hides his books under a loose floorboard. He checks the time and notices it's one in the morning, and it's now his birthday. Harry doesn't care much about his birthdays; he's never received a card. He walks to the window, wondering where Hedwig is.
The fact that Harry hasn't received a card before reinforces how alone he is, and not just because he's at the Dursleys': The house-elf Dobby kept him from receiving birthday cards last year, which though not Ron and Hermione's fault, certainly influences how Harry thinks of his birthdays and how others perceive them.
The narrator explains that the most unusual thing about Harry is the lightning bolt scar on his forehead. He got it twelve years ago when the dark lord Voldemort murdered Harry's parents, Lily and James, and failed to murder Harry. The curse rebounded onto Voldemort and he disappeared. Harry fought Voldemort last year, and he thinks he's lucky he made it to age thirteen.
For Harry, he recognizes that growing up is a privilege he's lucky to have, given how close he's come to Voldemort. His previous brushes with Voldemort suggest that the wizarding justice system isn't entirely effective, given that a child was more useful than the Ministry.
Harry notices three owls flying towards him and opens his window to let them in. The large gray owl, Errol, keels over. He belongs to the Weasleys and is old, so Harry unties the parcel and carries him to Hedwig's cage. Hedwig joins Errol once Harry takes her package. The third owl is from Hogwarts; it carries a letter and another package. Harry opens Errol's package and finds his first birthday card and a newspaper article saying that Mr. Weasley won the Daily Prophet's Grand Prize Galleon Draw. The accompanying photograph shows all nine Weasleys and Ron's pet rat, Scabbers, on vacation in Egypt. In a letter, Ron apologizes for the phone debacle and talks about the curses on the pyramids. He also incudes a note about his gift: a Pocket Sneakoscope, which is supposed to detect untrustworthy people.
The simple fact that Hermione and Ron send Harry gifts this year (Hedwig has Hermione's package) indicates that they're starting to grow up, become more mature, and discover different ways to show Harry that they care about him. With this, the novel situates itself as a midway point in the series that sees Harry, Ron, and Hermione transform from innocent and naïve children into functional adults, gradually becoming more mature as the series progresses.
Hedwig's parcel contains a card from Hermione. She writes that she's on vacation in France and asks if Harry will be able to meet her and Ron in London. Harry opens her gift, a broomstick servicing kit. He can't wait to use it on his Nimbus Two Thousand. Then, Harry picks up the final parcel. It's from Hagrid, the Hogwarts gamekeeper. Before Harry can get it unwrapped all the way, the object snaps like it has jaws. Harry picks up his lamp and then dumps the object out of the wrapping. It's a book titled The Monster Book of Monsters and it scuttles away.
Hagrid plays a special role in the trio's friendship, as he's both an adult authority figure and a friend who treats them like people worthy of trust and respect. Hagrid then becomes the trio's first indication that they can indeed be friends with adults. Hagrid's questionable gift, however, impresses upon Harry that Hagrid isn't a perfect friend, even if he is an adult.
Hoping the Dursleys are still asleep, Harry stalks the book and finally manages to tie a belt around it. He then opens Hagrid's card, which ominously says that the book will be useful for the school year. Harry turns to the letter from Hogwarts itself, which includes his booklist and a permission form that, if signed, allows the student to visit the village of Hogsmeade. Harry's heart sinks; he knows his aunt and uncle won't want to sign the form. He decides to worry about it tomorrow, crosses off another day on his countdown to September first, and lies down in bed to look at his birthday cards. He feels happy for the first time that it's his birthday.
The ominous note from Hagrid doesn't make sense to Harry now, but it does introduce the novel's exploration of perspective to the reader. In other words, what seems strange and ominous to Harry makes perfect sense to Hagrid, as this is the required textbook for his Care of Magical Creatures class. Hagrid also finds such items or creatures funny, while Harry clearly doesn't share this interpretation.