I must say that I thoroughly enjoyed playing these 'other world' zones in Dragonborn whereas I hated the similar 'other world' zones in Oblivion., second only to the stupid leveling system.
For a walk-through I am quoting from the UESP interspersed with my own pictures.
Chapter I
By reading the Black Book
Epistolary Acumen, you are transported to the realm of Hermaeus Mora, Apocrypha. If this is your first trip (not counting the one where you first meet Miraak),
Hermaeus Mora, who appears floating above the platform as a tentacled
eyeball, will then speak to you and tell of Apocrypha's wealth of
knowledge and lure towards those who seek it. Activate the scrye at the
center of the platform to open up a bridge over the sickly-green water.
Jump onto the moving passage, taking care not to fall in and continue
along. Hop off at the next platform to activate a second scrye and lower
a second bridge. Finally, use the moving platform to access the bridge
and read Chapter II, which lies on a pedestal.
Chapter II
Travel down the narrow pathway and activate another scrye. This opens
up the walls to create a larger platform for you to battle a lurker and a couple of seekers.
Once they are dealt with, activate a scrye off to the side in order to
open up the large gate across from it. Through it is Chapter III.
Chapter III
Eliminate the seekers in the room and continue onward. To the left,
there is a font of magicka surrounded by a pool of poisonous water that
will restore your magicka reserves. There is another one located at the
top of the stairs, where you will also encounter a moving passage
similar to the one encountered in Chapter I. If you choose to get off at
the closest platform, you will be greeted by a lurker and two seekers.
Drop down below for a large fighting space, then activate the scrye off
to the side and around a corner in order to return to the starting
platform. The last and farthest platform contains two seekers and a pod
full of loot with a font of magicka hanging on either side.
To reach Chapter IV, do not jump on the platforms. Instead, activate
the scrye at the end of the moving passage to open the gate. Defeat the
two seekers and continue to walk until you arrive at a lurker. At the
end of the path, you will need to activate yet another scrye to open the
gates in front of you. Use the font of stamina if you wish and continue
on, avoiding the tentacles that reach out at you from the water.
Eliminate the seekers as you walk down the path to your destination.
Chapter IV
A Daedric Prince wants to make a deal
You will recognize the location below the platform as the setting of
Chapter III. Loot the vessel to the left and activate the scrye. Walk up
to the Black Book: Epistolary Acumen and read it to summon Hermaeus
Mora. He will congratulate you and teach you the second word to Bend Will.
He continues to tell you that you cannot hope to surpass Miraak without
the last word of power, but offers it at a price: knowledge for
knowledge, he requests the Skaal shaman, Storn Crag-Strider. Once the conversation has ended, you may choose from one of three powers emanating from the Black Book:
- Dragonborn Flame — When your Fire Breath shout kills an enemy, a Fire Wyrm emerges from the corpse to fight for you for 60 seconds.
- Dragonborn Force — Your Unrelenting Force shout does more damage and may disintegrate enemies.
- Dragonborn Frost — Your Frost Breath shout encases foes in ice.
Only one ability from the book may be accepted at a time. To return
to Solstheim, use the Black Book in front of you or access it from your
inventory.
Next the Secrets of the Skaal.
