Professor Albus Dumbledore arrives at Privet Drive at the beginning of the first Harry Potter movie. This tall wizard of silver hair and beard puts out streetlamps using a Deluminator (Put-Outer) and addresses the cat in the corner, saying, "I should have known that you'd be here, Professor McGonagall." Turns out, McGonagall has transformed into her Animagus form (tabby cat) and she has been observing the Dursleys all day. She's on a first-name basis with Dumbledore and after greeting him, questions if the rumors are true. He confirms that both the good and the bad are true, and that Hagrid is bringing "the boy." Shortly after, the half-giant, Rubeus Hagrid, arrives with the infant on a flying motorbike and hands him over to Dumbledore. The truth gradually unfolds in the first Harry Potter movie and besides, there's a reason behind Dumbledore's decision to leave Harry on his aunt and uncle’s doorstep.
The trio walks towards the Dursleys' doorstep, and Minerva McGonagall seems particularly worried. She asks if it's safe to leave an infant with people who she thinks are "the worst sort of Muggles imaginable." Dumbledore is quick to remind her that the Dursleys are the only family of the infant. McGonagall nonetheless nudges Dumbledore to rethink, as the infant he carries is the sole surviving son of Lily and James Potter, after all. The Potters were great wizards who fought in the First Wizarding War as members of the Order of the Phoenix. The first chapter of Harry Potter and the Sorcerer's Stone "The Boy Who Lived" reveals that Lord Voldemort had turned up in Godric's Hollow the night before Harry was dropped off at Dursleys, and killed his parents. The infant was orphaned and while the Dark Lord vanquished, Harry Potter became The Boy Who Lived.
Petunia Was Jealous Of Lily
Harry Potter has had his name down at the Hogwarts School of Witchcraft and Wizardry ever since he was born. The sad part is, this boy knew nothing about magic or the nature of his parents' deaths until after midnight of July 31, 1991, when Hagrid arrived at the hut-on-the-rock. Harry lived a miserable existence in the cupboard under the stairs of Vernon and Petunia's home. Harry Potter and the Sorcerer's Stone makes it clear that Dumbledore's drop-off wasn't a goodbye, but a temporary arrangement. He felt that Harry was "far better off growing up away from all of that" and to that end, wrote a letter to the Dursleys and wished Harry luck. To answer the question of why Dumbledore decided to do so, it's pertinent to dive into Petunia Dursley (née Evans) and Lily's childhood.
Petunia was a Muggle and the elder daughter of Mr. and Mrs. Evans. Her younger sister, Lily, who was a Muggle-born witch, affectionately called her "Tuney." She, in turn, sneered at Lily and her friend, Severus Snape from Spinner's End as they had magical abilities and she did not. Petunia secretly longed to be a witch and wrote to Dumbledore as a child asking if she could be accepted to Hogwarts. Dumbledore politely declined due to her being a Muggle. Petunia resented Lily for being a witch and was particularly unhappy the day her younger sister left for Hogwarts. Chapter 33 of Harry Potter and the Deathly Hallows, "The Prince's Tale" reveals that the sisters had an altercation shortly before Lily boarded the Hogwarts Express at Platform Nine and Three-Quarters. Petunia called Hogwarts "a stupid castle" and her sister "a freak" while wishing she too were boarding the train. Having had enough, Lily confronted Petunia about writing to Hogwarts' headmaster, and their relationship turned sour.
Petunia was jealous of Harry's magical heritage, and she confined him to a cupboard under the stairs. Harry Potter and the Sorcerer's Stone shows cousin Dudley's attempt to cause discomfort to Harry by stomping up and down these stairs. The Dursleys did their best to intercept, and destroy the first and subsequent Hogwarts letters addressed to "Mr. H. Potter, The Cupboard under the Stairs, 4. Privet Drive, Little Whinging, SURREY." Harry wore Dudley's hand-me-downs, cooked breakfast for the family, and served Uncle Vernon coffee and biscuits. Vernon was cruel, Petunia was harsh and Dudley was a bully. So, the question remains as to why Aunt Petunia became Harry's guardian.
Dumbledore Wrote To Petunia
Chapter 37 of Harry Potter and the Order of the Phoenix "The Lost Prophecy" answers this question, via the conversation between Albus Dumbledore and a raging, Harry Potter. After back-and-forth about Snape, Sirius, and the Black family house-elf, Kreacher, Dumbledore addresses the question of leaving Harry on his aunt and uncle's doorstep. He reveals that he'd planned and intended for the young wizard to arrive at Hogwarts, safe, and admits that he knew the young boy would experience ten difficult years at Dursleys. "You might ask — and with good reason — why it had to be so. Why could some Wizarding family not have taken you in?" remarks Dumbledore as Harry listens. The Wizarding families would have gladly taken Harry in and raised him as a son, so why the stingy Dursleys, after all?
In Harry Potter and the Order of the Phoenix book, Dumbledore reveals that his top priority was to keep James and Lily's son alive from the great danger he was in. Though Voldemort had disappeared, his Death Eaters were baying for blood. Dumbledore was sure that Voldemort would come back and knowing that the Dark Lord's knowledge of magic was more extensive than any wizard alive, he made the decision about infant Harry's years ahead. Dumbledore also knew that Voldemort wouldn't rest until he killed Harry Potter. "I knew that even my most complex and powerful protective spells and charms were unlikely to be invincible if he ever returned to full power," says Dumbledore. He also knew where Voldemort's weakness lay, in that, Harry would be protected by "an ancient magic" that Voldemort knew of, despised, and underestimated. Dumbledore addresses the fact that Harry's mother died to protect her infant son and that she gave him a lingering protection that Lord Voldemort never expected. "I put my trust, therefore, in your mother’s blood. I delivered you to her sister, her only remaining relative," he explains.
When Harry says Aunt Petunia never really loved him, Albus Dumbledore explains that by taking him in, she sealed the charm Dumbledore placed upon him. "You need return there only once a year, but as long as you can still call it home, there he cannot hurt you," explains Dumbledore. Simply put, Harry couldn't be harmed by the Dark Lord at the Dursley household because it was the place where his mother's blood dwelled. Chapter 37 of Harry Potter and the Order of the Phoenix reveals Dumbledore himself explained what he'd done in the letter that he left with Harry on Petunia's doorstep. She was aware that allowing her nephew houseroom would keep him alive. Harry had lived for the last fifteen years (ten before Hogwarts and five from his arrival to this confrontation). Petunia and Vernon weren't kind caretakers (as seen in Harry Potter) but they did agree to take Harry in just to keep him alive. While yes, Petunia despised her late sister and mistreated her only living child, she still kept Harry out of a sense of familial duty.