There was a king named Fornjot, he ruled over those lands which are called inland and Kvenland; that is to the east of that bight of the sea which goes northward to meet Gandvik; that we call the Helsingbight. Fornjot had three sons; one was named Hler, whom we call Ægir, the second Logi, the third Kari; he was the father of Frost, the father of Snow the old, his son's name was Thorri; he (Thorri) had two sons, one was named Norr and the other Gorr; his daughter's name was Goi.
Norr and Gorr end up conquering everything, and they split the winnings between them: Norr rules the mainland (hence the name 'Norway'), and Gorr the sea. Gorr the Sea King has two sons, Heiti and Beiti. We then learn that Heiti is the father of Sveiði, who is the father Halfdan the Old, who is the father of Ivar, who is the father of Eystein the Clatterer, who is the father of the wise Earl Rognvald. Rognvald had a son, the Hrolfr who won Normandy, also known as Göngu-Hrólfr (which literally means Walking Rolf, a nickname he supposedly got because he was too big to ride a horse). This Hrolfr is more commonly known as Rollo, who is the great-great-great-grandfather of William the Conqueror. Tracing genealogies back that far always involves a considerable number of assumptions, but Rollo shows up in my family tree multiple times, so I'm at least fairly certainly descended from multiple Norman families who were certain they were descended from Rollo, which is about as far as you can get in talking about genealogical connection to a person in the tenth century.
While the genealogy may look fairly mundane, it is not. Fornjótr, king of Gotland, Kænland and Finnland, is a jötun, a giant. Aegir is the sea; Logi is fire; Kári is wind or storm. These are all explicitly said elswhere. And, of course, Kári's son Frosti (frost) has a son named Snaer (snow), who has a son named Thorri (deep winter or frozen snow). Sacrifices were made to Thorri (or maybe he's invented to explain the sacrifices by that name), and a month shares his name (roughly from mid January to mid February). He's sometimes thought to have originally been Thor, although I've never seen any argument for that beyond the similarity of the names. Gói, the sister of Nórr and Górr, is light snow, and is also the name of a month. With them the divine age is starting to shift into the heroic age, and with Rögnvaldr the Wise, an associate of King Harald Fair-Hair, the age of heroes is morphing into the age of men. So now you know why I'm so awesome; I'm descended from Storm, Frost, Snow, and Winter. Of course, by this point a very large number of people with Norman and Scandinavian roots could say the same; it is no secret that Scandinavians are the children of Snow and Winter.
This descent-from-the-gods is usually explained as an act of propaganda, families trying to establish their legitimacy and so making up something that makes them special. Forging genealogical connections is not unheard of, so I've no doubt that there have been such cases. But if you look at the contexts, this is not a very good explanation of most of the evidence. For one thing, the ones that look most fictional are obviously allegorical; we don't know in exactly what way they originally took them. We tend to think of genealogy as a description of biological descent; it's very clear that this is not the only function genealogy has served. It is arguably not even the primary function of it through most of history; there is some reason at least to think that people often saw genealogy as symbols, as descriptions of themselves. There's some reason, similarly, to think that people are often not just making up the connections, even many of the most fictional ones; they are discovering connections that they think can be plausibly understood as genealogical -- perhaps such-and-such source says that so-and-so went to such-and-such place and married; perhaps a name loosely like so-and-so's runs in a local family there; perhaps they take the speculative connection as sufficiently established for their purposes. Or another case: Everyone, we are told, is descended from Adam; so where's the point at which we most likely connect up to his family tree? It's bound to be somewhere.
There's also remarkably little interest in exclusivity in most cases; there's no attempt to claim that the family in question is the only family to descend from the god, or even that it's the most important. It's rather like visions of past lives, in which everyone turns out to have been Cleopatra. There are lots and lots of families descended from Fornjótr. By the time of the Orkneyinga Saga, being descended from Fornjótr would not have been a rare distinction; if it can be claimed of Rollo in one line, it can be claimed of many people in many different lines. What is more, these genealogies are often serving not to single out families as special but to situate them in a broader scheme of human life. Many of the older examples are pretty clearly not dynastic genealogies but tribal or national genealogies. The kings and nobles are just the eminent examples. Beowulf is not marking out the Scyldings as special Danes; it doesn't make a sharp distinction between Scyldings and non-Scyldings at all. I plausibly have a Frankish noble house or two in my family tree, as well, so I am assured by certain Frankish sources that I am descended from Helenus, son of Priam of Troy, and thus ultimately from Zeus. That's not surprising, though; the really surprising thing would be to find someone with any Frankish ancestors who isn't.
What is more, many of the genealogies we have were not put forward by people who believed in the gods. Jordanes traces the Gothic royal houses to the Aesir; he didn't believe that the Aesir were divine, but the Romans had gods in their genealogy, so why wouldn't Gothic kings? We often have no evidence at all, even indirect, of a descent-from-the-gods genealogy prior to the coming of Christianity; most of them were compiled by Christians from literary sources. And it is notable that the Norse genealogies tend to treat the whole thing in a matter-of-fact way; there's often no attempt to flag the fact that someone is a god or giant unless it matters to the story, and sometimes the genealogist is going well out of his way to make someone not a god. Euhemerist assumptions can partly explain this; but it's hard to shake the feeling that the people put the gods in not because they were gods but because they were old, that it was not special roots but deep ones that mattered. The Norse genealogies sometimes trace families back not only to Norse gods but to Greek ones as well, as if to say to Greek and Roman civilization, "We're not less deeply rooted than you." If you look at how people do genealogy today, something like that idea is occasionally visible. And if that was the idea, they certainly were not wrong to that extent.