WARNING: The following contains spoilers for Episode 15 of Yashahime: Princess Half-Demon, "Farewell Under the Lunar Eclipse" now streaming on Crunchyroll, Funimation, Animelab and Hulu.
The mystery of what happened to Inuyasha and Kagome has haunted Yashahime: Princess Half-Demon from its very first episode. 15 episodes in, the series finally reveals the couple's fate -- though it would seem their story isn't over just yet.
Episode 15's action revolves around Kirinmaru. A deadly fragment of the Grim Comet is descending, which in the past he and the Great Dog Demon dispelled. This time, Sesshomaru and Inuyasha elect to deal with it, as they don't want to wake Kirinmaru. He's deadly and loathes all humans, so they decide they're better off without him. The comet fragment does not turn out to be too much of a threat on its own -- Inuyasha uses his Meido Zangetsuha attack to easily dispel it, seemingly solving the problem. Of course, by this point, Kirinmaru has been awakened, and Inuyasha is already in a world of trouble.
The mystery demon, Zero, is the one at fault here, and a major player in the events of this episode. She initially threatens Sesshomaru, mocking him for having half-demon daughters and warning him to hide them, as Kirinmaru will certainly try and murder them. She also sends Joka to attack Sesshomaru, though this obviously doesn't go particularly well. Most importantly, she is the one who wakes Kirinmaru. She wakes him to warn him of the possibility of a prophecy being fulfilled-- one that would mean his downfall.
Kirinmaru, as it turns out, has a very particular reason to desire the half-demon princesses' deaths. It was prophesied that he would ultimately be killed by neither a demon nor a human. Towa and Setsuna are half-demons, and Moroha is a quarter-demon, so they might very well be fated to confront and kill Kirinmaru.
This episode also reveals how Towa and Setsuna wound up in the forest, and that their mother was Rin. Sesshomaru is clearly not the most communicative of husbands, however, as immediately after Towa and Setsuna are born, he whisks them off to the forest. He knows that Kirinmaru will kill them, so he puts them under a spell of protection. In a somewhat comical moment, he chops off Joka's arms and steals her gold and silver rainbow pearls with complete nonchalance, gifting them to his new daughters. Viewers don't see what happens to Rin after this, so it remains to be seen how she became trapped in the tree of ages, but it can be assumed Sesshomaru has something to do with it.
When Kirinmaru is about to kill Inuyasha and Kagome to take Moroha, Sesshomaru rushes forward and seals the two within the black pearl. He does this because, on the surface, there will be a time when the Meido Zangetsuha ability is needed again. In reality, however, Sesshomaru is likely protecting Inuyasha and Kagome the best way he knows how. At any rate, it's something of a relief to know that neither of them is dead, though who knows what kind of life they're living inside the black pearl.
Characters being magically sealed away is quite common on Inuyasha, but Inuyasha and Kagome's imprisonment is still pretty tragic. They've missed out on raising their daughter, and even if they're released, nothing can change that. The same goes for Rin -- though even as Sesshomaru takes away their daughters, she says she trusts him.
Finally, it's worth mentioning Riku's involvement in all this. Riku has been working behind the scenes, manipulating events for a long time, it seems. He's really the reason Inuyasha and Kagome get the Black Pearl and the Beni, and so he's directly responsible for their fate in many ways. He's certainly planning something big.