To be fair to English history, there are more instances of not marrying first cousins (particularly before the eighteenth century), but still, to our modern sensibilities, being able to trace back a romantic prospect to a shared ancestor of any kind is creepy. How many people have you heard point out that Eleanor and Franklin Roosevelt were related (fifth cousins, but both in the male line, so Eleanor's maiden name was also Roosevelt) with a tone of horror?

Historically, royal marriages involved more than just happening to fall in love with someone. As I discussed in this previous answer, a certain level of equal prestige between the parties was important. This could seriously limit potential partners. Some of this could be about pride, not wanting to "dilute the royal blood" with a relationship with a minor noble or an untitled subject; some of this related to princesses serving as unofficial ambassadors in foreign courts, as I described in the earlier answer, and royal parents not seeing a good reason to waste their children in marriages where they would not be able to act as diplomats; some of this was also financial, as the crown could be obligated to support the family of a princess who married a noble or commoner instead of into a position at a foreign court. All of this meant that the pool of possible spouses was sometimes low, which might sometimes push for a closer alliance with a cousin. At the same time, the modern political map does not come close to representing the number of kingdoms, princedoms, and important duchies in pre-modern Europe, and British/English monarchs made a lot of varied choices for themselves and their children. Prior to George I, English monarchs married into families across Europe, particularly with Spain, France, and important Spanish and French duchies.

The only first-cousin marriages I know of in the various English/British royal families are are George IV and Charlotte of Brunswick-Wolfenbüttel (shared grandfather: Frederick, Prince of Wales), and Victoria and Albert of Saxe-Coburg and Gotha (shared grandfather: Francis, Duke of Saxe-Coburg-Saalfeld). Not that many, despite the stereotype! George V and Mary of Teck were second cousins once removed (shared (great-)great-grandfather: George III), and then there were probably a number of third cousins, like Elizabeth II and Philip of Greece and Denmark (shared great-great-grandmother: Victoria), and Edward I and Eleanor of Castile (shared great-great-grandfather: Henry II), but they are more difficult to trace. Given the pressures described above regarding the size of the pool of potential royal spouses, it's not surprising that second and third cousins would turn up with some frequency. But let's look at the two first-cousin marriages and see what made them happen.

George IV (then Prince of Wales) and Charlotte of Brunswick-Wolfenbüttel (1794)

George, the oldest son of George III, was, at this point, mentally if not actually married. Maria Fitzherbert, a beautiful and wealthy double-widow, had been his mistress for ten years; they had engaged in a secret wedding ceremony that was not legally binding, since the Royal Marriages Act of 1772 made the reigning monarch's consent a requirement for the weddings of members of the royal family, but they certainly seemed to consider it "real". (He was eventually buried in 1830 with a miniature portrait of her eye (it was a thing) and she wore widow's weeds, despite their long estrangement.)

While George not having any children would not have led to a succession crisis, as he had a ton of brothers that could have succeeded him, several of his younger brothers were also involved with invalid marriages and basically were not on track to be respectable royal dukes and potential kings. His father therefore worked on him to get married-for-real first - it's just logical to start from the top. The Hanoverian kings (Georges I, II, and III) preferred to bring brides in from various German states, and to send their daughters there; George I was German, and his and his descendants' family circle effectively stayed German, so it was natural to look for a German princess for the prince of Wales. This, of course, cut the pool of potential brides way down - there were not many unmarried Protestant German princesses of the right age to get married in 1794. Caroline also looked great on paper: she was attractive, she was clever, she was a princess, and her mother had been an English princess on good terms with his father (who, after all, was masterminding this). It would transpire that the marriage wouldn't work out, George being boorish and in love with someone else and Caroline not being the sort of person who would deal with that well, but it seemed like a really good idea at the time.

Victoria and Albert of Saxe-Coburg and Gotha (1840)

Unlike George IV, Victoria was in full control of her own marriage, as she was the reigning monarch at the time. She was, however, still under pressure to get married relatively soon to assure the succession. Her maternal uncle, Leopold of Saxe-Coburg-Saalfeld, King of the Belgians, had once been married to her paternal cousin, Princess Charlotte, daughter of the couple in the section above. (She died very young.) Leopold had taken an interest in Victoria when she was still a princess, and made an effort to have her meet his and her mother's mutual nephew, Albert, as a potential spouse. His efforts were successful, and she kept Albert in mind for several years and eventually married him.

Albert's good looks played a part in Victoria's attraction to him; he was also Protestant, and of lowly-enough birth that he wouldn't overshadow her or appear her equal. As with Caroline, there were decent on-paper reasons for the alliance. And first cousins marrying at this time was just not seen as incestuous or creepy - modern disgust for cousins marrying seems to date back to the very late nineteenth century, and even with that, it's not illegal in many places.