Part of it is nowadays, you expect to marry freely with minimal requirements. Not in a certain group (endogamy), or up or down (hypergamy for the one marrying up), or across (isogamy). Requirements were rather sterner in older days.
Fairy tales tend to treat marrying up with casual abandon, with peasant's daughters, woodcutter's sons, old soldiers, merely noble daughters, managing to marry kings or princes or princesses. Though there's an entire group of fairy tales where there is a prophecy that a poor man's son will marry a king's daughter, or sometimes just a merchant's, and the father-in-law-to-be goes to great efforts to prevent it. And of course, the king is prone to react to the stupid third son of some peasant by throwing up more obstacles. To be sure, they always fail and are always treated as rather like fools. . . though in literary fairy tales, it can actually pull more weight. The Facetious Nights of Straparola has a tale where a late-born princess, in spite of having no land dowry because her father the king had divided his estate between his three older daughters as their dowry before she was born, nevertheless assures him that she will never disgrace her blood.
Next - speaking with all submission and reverence - I do not purpose to let myself fall be low the race of my ancestors, who from all time have been famous and illustrious, nor do I wish to debase the crown you wear by taking for a husband one who is our inferior. You, my beloved father, have begotten four daughters, of whom you have married three in the most honourable fashion to three mighty kings, giving with them great store of gold and wide domains, but you wish to dispose of me, who have ever been obedient to you and observant of your precepts, in an ignoble alliance. Wherefore I tell you, to end my speech, that I will never take a husband unless I can be mated, like my three sisters, to a king of a rank that is my due
Well, that was only one tale, to be sure. In others, people marry up. Madame d'Aulnoy was made of sterner stuff. Even when she was retelling a fairy tale like The Dancing Water, the Singing Apple, and the Speaking Bird (as Princess Belle-Etoile) or combining Cinderella and Hop O'My Thumb (as Finette Cendron), where the heroine had to be in a lowly position up front, she figured out ways to make the poverty-stricken family royal.
Madame d'Aulony also had a surprising tendency to make them first cousins. Though not as impressively as the first author of Beauty and the Beast, who made Beauty and the Beast double first cousins. To be sure, being royal, they could probably have gotten dispensations. When the Church was promulgating very wide restrictions on the marriages of cousins, one French king was praised for having obeyed it by the expedient of lowering himself to marry a mere noblewoman. In ancien regime times and the nineteenth century, that would have gotten a colder treatment. Queen Victoria got complaints that she allowed princes and princesses to marry the offspring of morgantic marriages, and would scold back that if she thought them appropriate, who was anyone else to disagree? (She even let a daughter marry a subject.) Then, one thing she also wrote about was the immense desirability of getting some brown-eyed blood into the blue-eyed royal family.
Inbreeding was, after all, a known consequence of closer intermarriage. Though the dangers are much exaggerated. Even closer than cousins, most children turned out all right, and many cultures swing it. Plato's Laws take for granted that any heiress will be rapidly married off to her nearest male cousin, and when it's occasional, it can be fine. It's intensive, continual intermarriage over generations that really brings out the genetic problems.
Then, some writers treat it as the sole danger, so all incest taboos are based on it. This is obviously impossible because relationships are frequently not prohibited on the degree of consanguinity. Many cultures forbid parallel cousins (children of brothers or sisters) from marrying -- no matter how distantly, in China for instance you can't marry anyone of the same last name -- while actually preferring cross cousins (children of a brother and sister). There are also forms of incest that do not involve consanguinity at all. Affinity is the commonest; you may not be able to marry your stepparent (or child), your sibling-in-law, your parent (or child) in law. Puritan New England argued, from "one flesh" that a married couple were now related to each other's family exactly as the spouse had been, and so marrying your dead wife's cousin was no different from marrying your own cousin (And not just for marriage. One women was sent a letter with greetings for "your brother John" -- where John was her husband's dead first wife's sister's husband.)
Saint Augustine thought that it sprang from the need to force people into new relationships. By marrying a woman who was not his sister, a man would acquire a father-in-law he wasn't related to, instead of merely making his father his father-in-law as well. (Yes, that was the relationship he suggested. Tells you something about family relations back then.) He certainly has a point that incest taboos serve to structure society, especially in cultures where ties of blood and marriage matter.
Other cultures had other requirements. Castes in India might have other castes, frequently inferior, from which they choose their brides. British writers reported that some castes that killed their daughters did so to avoid marrying them off, sometimes just for expense, but sometimes because they would have to marry down. European royal daughters frequently had to remain unmarried because their fathers were unable to find a suitable equal marriage. And many impoverished patricians went to marry off their sons to the wealthy upstart's daughters, rather than the other way round. (Though it was not, to be sure, unknown. One study of German men raised to the nobility during the nineteenth century found that most had either a noble wife or a noble mother.)