How do I activate the dwarven item in Raldbthar?

I quote the Elder Scrolls Wikia:

At the end of Raldbthar there is a Dwarven Mechanism which you lack the necessary item to activate. (First Dark Brotherhood Contract Area For Muiri)

The item is an attunement sphere found in other dwemer ruins. To unlock the mechanism you need to travel north of Winterhold. Carefully walk down the mountain, or jump off the bridge path to the college and pray you land in the water (not suggested) then swim north past Skytemple ruins, to a cave called Septimus Signus's Outpost. Once you have talked to Septimus he will give you a quest called Discerning the Transmundane and give you key to open the Dwarven Mechanisms.

Raldbthar connects to Blackreach (by going through the Mechanism) which is part of the Discerning the Transmundane quest, so if you dont want to explore the location Septumus suggests or if you already went through that gigantic location and dont want to retun yet, then you can ignore the mechanism.

TES:V Wikia

Also for me I went there and got to the end where the mechanism is but I couldn't open it so I left. I got the piece and went back but at the end where you pull out the junk from the gears, there wasn't any pieces to get unstuck{the junk in the gears wasn't there since I already pulled them out} so I would wait until you have the piece before completing the dungeon.

There is a back entrance you should have unlocked. If you go to where the first entrance is but keep going up instead of in there is a tower-like structure with a dwemer elevator leading you right to the area with the contraption. That way you don't have to walk all the way through. This of course requires that you have opened that alternate entrance/exit from the inside to begin with however. Always make sure to open backdoor shortcuts in dungeons when they are available in case you need them for later quests to to pick up chests etc. That you couldn't get at that time.

the last piece stuck in the gears is underwater. it connects to the one thats stuck, just follow the pipe leading down the stairs. you dont need the last peice in order to pass through.