Why can't I return the Golden Claw to its owner?

Solution 1:

There is a wall near the coffin where a word is glowing. You have to walk close to it, the screen will get dark until only the word is visible. You'll then receive a message saying that you acquired a new shout.

Then, the boss will leave the coffin behind you and start to attack you. Kill him and make sure to loot his corpse if you don't want to come back later. You can now exit the cave and return the claw.

Solution 2:

You first need to absorb the runes that light up. Then, you need to walk to the coffin, that is close to the runes that light up (it should be marked by the quest marker if I remember correctly). The boss will leave his coffin and will start to battle you. Once he is dead, you can return to finish your quest and continue.