Short answer: Victoria inherited the throne because succession law in the UK preferred the children of older sons to younger sons. Male children had priority, but female children would inherit in the absence of any surviving male siblings. This doesn't trump male-preference primogeniture; it is how male-preference primogeniture works. The male preference governs the choice between siblings: a younger brother inherits before his older sister, but in either case the child of an older brother inherits before a younger brother.

(As background, the reason Ernest Augustus inherited the Kingdom of Hanover is because Hanover used the Salic law, which was established under Clovis, the King of the Franks, around the turn of the sixth century. This includes the key phrase "But of Salic land no portion of the inheritance shall come to a woman: but the whole inheritance of the land shall come to the male sex." This is not to say that all of the Salic law was still in effect in 1837, but that this particular tenet still governed succession in Hanover.)

The UK's law preferred Victoria, a woman born of an older brother (Edward, Duke of Kent and Strathearn, who was himself never king because he died before his older brothers) to any surviving younger brother. The reason is that British succession law was what is called male-preference primogeniture succession. (That law has sinced changed, in 2015.) By this law, the right to inherit was passed down through oldest male children to their children. Younger brothers would inherit over older sisters, but in the absence of a son the daughter would inherit before her father's younger brothers or their children. This meant that the heir apparent of the reigning monarch was:

the monarch's oldest son,
if he was dead, then the oldest living son of the oldest son,
if the monarch's oldest son didn't have a living son, the oldest son's oldest daughter,
if the oldest son died and had no surviving children, the next-oldest son,
if he was dead, the oldest son of the next-oldest son,
if there wasn't one the oldest daughter of the next-oldest son,
and next in line was the third-oldest son,
etc.

I left out grandchildren, but they follow the same rule: you trace down descent through the oldest male child before going on to the next-oldest for succession. I feel like this answers your question, but if you're curious, you can read a little more about the actual case of Victoria, below. :)

In the case of the children and grandchildren of George III, the succession would have passed to Charlotte, Princess of Wales, the only legitimate child of the Prince Regent, later George IV. After his death, she would have become queen, even with George's younger brothers surviving. The House of Hanover in the early 19th century was kind of a mess, but Charlotte's kindness and gentleness was seen as a ray of hope after her insane grandfather and famously gluttonous, wasteful, and lecherous father. Her father wanted Charlotte to marry William the Prince of Orange, but Charlotte hated him and refused (she wrote in a latter that, if she were forced into the marriage, she wouldn't leave England to stay with him: "Therefore the P of O must visit his frogs solo". I love this detail so much I had to include it). She was in love with a dashing young Prussian whose identity is unclear. Unfortunately, the match wasn't to be, and Charlotte married Prince Leopold, later King of Belgium. When Charlotte became pregnant, the kingdom celebrated. The royal line seemed assured. The Prince Regent had one heir, but she was a good one and loved by the people, and she was about to have a family of her own. Then, tragedy struck. Charlotte had been weakened throughout her pregnancy by bad medical care. After a long and painful labor, she delivered a stillborn child and died shortly after. What had been a cause of joy plunged the country into mourning. Her stillborn child was a son. This was in 1817.

At this point, George III's sons scrambled to have legitimate children, since suddenly one of them or their child would inherit. (Out of fairness, I should say that unlike his older brothers Ernest Augustus was already married and, it seems, actually faithful to his wife. The others had to either get married quick or try to patch up their marriages.) The oldest to have a legitimate child would ensure the new line. Let's take a look at the field. Of the children of George III, at this point in 1817 there was, starting with the oldest:

*George, the Prince Regent, born 1762, who had no surviving children after Charlotte's tragic death. He became George IV after his father's death in 1820 and died in 1830.
*Prince Frederick, born 1763, married but estranged from his wife and with no children. He would have succeeded his brother had he outlived him, but died in 1827.
*Prince William, born 1765, married but with no legitimate children. He had scads of illegitimate children, but those don't count. Succeeded as William IV in 1830 after the death of George IV his brother. Died 1837.
*Charlotte, Princess Royal (not to be confused with Charlotte, Princess of Wales), born 1766, died 1827, and had no surviving children. In order for her to inherit the throne, every single one of her brothers and their children (and any grandchildren) would have had to have died.
*Prince Edward, Duke of Kent and Strathearn, born 1767. As the oldest son of George III to have a surviving child after 1817, he won the race, so to speak. He had a daughter named Alexandrina Victoria, born 1819. Had he had any sons, the oldest son would have inherited the throne. Edward would have succeeded to the throne had he outlived his older brothers, but he died in 1820.
At this point, since an older brother's claim passes to his children before his younger brothers, the younger brothers no longer count. Sorry, chaps. Should have been born sooner.

George (later George V of Hanover), son of Ernest Augustus (the next-oldest son of George III, born 1771), was born three days after Alexandrina Victoria. If she hadn't been born or had died before 1837, Ernest Augustus would have come to the throne of both the UK and Hanover, and after his death the UK would have had a George V rather sooner than they actually did. There were rumors that her uncles tried to have Alexandrina Victoria assassinated before she could take the throne, but whether or not that's true, she lived.

Thus, if you're still with me, we arrive in 1837. Alexandrina Victoria is now (barely) 18 years old. Her father has been dead since 1820, and her father's last remaining older brother William has just died. As I'm sure you all have already guessed, as her father's older brothers have no surviving legitimate children at this point and her own father is dead, she becomes Queen Victoria.