Ascend the stairs and you'll come out on the balcony level of the previous room, where you can find several dead draugr and the corpse of Daynas Valen himself. Don't disturb anything just yet, though- look to your left and you'll see the walkway that is dropped by the keyhole Daynas is leaning on. A careful peer between the slats will show that there are at least two draugr over there waiting for you as well- not a welcoming comittee that you want to encounter by surprise at all. If you've got the ranged ability, go ahead and find a way to pick them off before lowering the bridge. If not, well, you'll just have to make ready. Though, if you have no long-range capability at all, you're likely to be too low-level to be in here, so keep both hands on the seat of your pants in case something comes along with a large boot.
Looting Daynas will turn up several interesting items, including not only the Ivory Dragon Claw, but also a book of Daynas' notes. You will need to read this to continue the quest.
According to Daynas' notes, there was an archmage by the name of Gauldur. He was well-liked and powerful, and respected by all. His three sons, though, always lived in his shadow and became steadily more unhappy with this. Over time their thirst for a piece- preferably every piece- of their father's renown and strength led the eldest to discover an amulet that Gauldur always wore. Supposedly, this amulet was the source of his might and wisdom, so the sons of Gauldur killed him in his sleep, taking the amulet and dividing it into three parts so that each could have one.
Of course, being of the sort to kill their father meant they had no scruples about destroying nearby villages and taking all of their stuff, so the High King sent battlemages led by Archmage Geirmund to put an end to the destruction. While the brothers were defeated they were not killed, and fled for their lives.
The youngest, Mikrul, was trapped here in Folgunthur, and after three full days of battle he was finally defeated and entombed beneath the lock of the Ivory Dragon Claw.
Jyrik, the second of the sons, was chased to Saarthal, where he was able to slay ten of the attacking battlemages before he too fell, and was sealed there.