To understand why Britain adopted a German royal family in 1714, you need a bit of background on some political and religious events of the 17th century.

First, the Glorious Revolution of 1688: Britain's king James II (a Stuart) had squandered a lot of goodwill by acting in authoritarian ways--he overturned charters that governed Britain's colonies such as Massachusetts Bay; he tried to start a war with the Dutch that no one else in England wanted; he continuously prorogued (discontinued a session of) Parliament so that it could not meet. But by far the number one cause of his unpopularity was a religious one: James was a Catholic. In a nation whose identity was extremely tied to its Protestant faith, this was a huge deal. James also had a son who would be brought up as a Catholic himself, which meant that the English crown would remain a Catholic one in the future. (As a side note, Britons weren't only worried about the king being a Catholic: they also feared that he would force Catholicism on the nation as a whole, like Mary Tudor had attempted to do in the 16th century.) Parliament invited William of Orange to invade and become monarch in large part because William was a Protestant prince, and his rule would therefore ensure that the English crown would be held by a family of Protestants.

Now, unfortunately for Britain's Protestants, William and Mary died childless, and in 1714, Mary's sister Anne also died without an heir. Which put Britain in the same dilemma that they had been in in 1688: where could they find a Protestant monarch? In 1701, Parliament had passed an Act of Settlement decreeing that Britain's monarch must be a Protestant, so this was a legal requirement at this point. To find a Protestant monarch, they searched the Stuart royal family for the nearest Protestant relative. And that's how they got to Georg Ludwig, Elector of Hanover. He was not the closest relative to the previous queen, Anne Stuart, but he was her closest living Protestant relative of Stuart descent. In determining the line of succession, Parliament dismissed fifty claimants with closer hereditary claims because of their Roman Catholic faith.

By the way, not all Britons were thrilled that a German had been brought in to be their monarch. In fact, the seventeenth and eighteenth centuries were plagued by Jacobite uprisings: attempts to replace the Hanoverian royal family with James II, his son, and grandson, who were living in exile in France and Italy. These uprisings were especially common in Scotland, where the Stuarts had always been more popular. (This popularity was in part because the House of Stuart was originally a Scottish royal family.) In one of these attempts, the Jacobite Rising of 1745, the Jacobite forces led by James II's grandson even managed to capture the Scottish capital of Edinburgh for a period of time. By the time of George III's accession to the throne in 1760, however, the House of Hanover had become widely accepted by the majority of the British populace.

In sum, the German Hanoverians became British monarchs because they were Protestants. However, not all Britons were thrilled by this choice, leading to a fair measure of dynastic upheaval.

Sources:

Steven Pincus, 1688: The First Modern Revolution (2009)

Allan Macinnes and Brent Sirota, eds., The Hanoverian Succession in Great Britain and its Empire (2019)