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In the 3rd movie (Harry Potter and the prisoner of Azkaban), Harry left his aunt's house after blowing up Marge and waited in the street. After a few minutes, a bus called the Knight Bus arrived and took harry to a place that Harry called Leaky Cauldron.
How did the bus get to Harry and why did Harry ask to be taken to that place?
Bonus question, why were Hermione and Ron there?
Harry inadvertently summoned the bus by waving his arm around.
Harry stepped backwards. His legs hit his trunk and he tripped. His wand flew out of his hand as he flung out an arm to break his fall, and he landed, hard, in the gutter.
HP and the Prisoner of Azkaban
and
For witches and wizards who are Floo-sick, whose Apparition is unreliable, who hate heights or who feel frightened or queasy taking Portkeys, there is always the Knight Bus, which appears whenever a witch or wizard in urgent need of transportation sticks out their wand arm at the kerb.
and
’Ere,’ he said, looking suspicious again, ‘you did flag us down, dincha? Stuck out your wand ’and, dincha?’
‘Yes,’ said Harry quickly. ‘Listen, how much would it be to get to London?’
Harry Potter and the Prisoner of Azkaban - Chapter 3: The Knight Bus
Note that in the novel his goal wasn't to get to the Leaky Cauldron, it was to get to his money in Diagon Alley.
They were thundering along Charing Cross Road. Harry sat up and watched buildings and benches squeezing themselves out of the Knight Bus’s way. The sky was getting a little lighter. He would lie low for a couple of hours, go to Gringotts the moment it opened, then set off – where, he didn’t know.
Harry Potter and the Prisoner of Azkaban - Chapter 3: The Knight Bus
Whereas in the film he simply names somewhere in London that he knows will be amenable to magic folk.
STAN SHUNPIKE: Whereabouts you headin'?
[Harry hesitates. He hadn't thought about this. Decides.]
HARRY: The Leaky Cauldron. That's in London --
Harry inadvertently called the bus by holding up his wand. You could say it's a plot convenience that it was nearby or it just teleported where it was needed.
Leaky Cauldron is a wizarding pub and inn. Behind it is a secret entrance to Diagon Alley. Hagrid visited it with Harry in the first book/movie. It's where they first met Professor Quirrell. Harry named it because he knew he could spend a night there. Diagon Alley is the closest magical place he knew about, so he went there.
Hermione and the Weasley family joined him much later, when it was time to go to Hogwarts.
When Harry blows up Aunt Marge, he packs his trunk and walks out, fuming with anger, upto Magnolia Crescent. There he sees the silhouette of a giant black spectral dog (Sirius) and points his wand at him. While doing so he is positioned probably at the edge of the road's sidewalk and has his wand hand sticking out (as shown in the film). By doing so he unintentionally summons the Knight Bus. Harry's head was probably very cluttered with the ensuing events and Leaky Cauldron might have been the first safest place he thought of going to. Ron and his family were there after their visit to Egypt. Hermione might have been in contact with the Weasleys and so decided to meet them before returning to Hogwarts. Fudge knew where Harry was. He might have informed Dumbledore. Knowing Harry's friendship with Ron Dumbledore might have requested Arthur Weasley to keep an eye on Harry's actions. P.S. This answer is a lot of guesswork.
Well, if you were a true Potterhead you would know, after losing control of his magic and blowing up Aunt Marge, he packs up and storms off up Magnolia Cresent, were mysterious things begin to happen to the park equipment, then the Animagus form of Sirius Black creeps out of the bush to stare at Harry, worried, Harry stands up onto the side of the road and flings his wand up, unintentionally summoning the Knight Bus, were he meets somewhat annoying conductor Stan Shunpike, and the crazy bus driver Ernie Prang (fun fact, JK Rowling chose the names Stan and Ernie because they were her grandparents names!) Anyway, Harry aboard the Knight Bus And has and eventful trip to the leaky cauldron were he meets Fudge, and later meets the Weasleys and Hermione to go to Hogwarts.
