The only available answer is "Because the Ambitions are people and should get to make their own decisions."

Boy, can you tell that my character has formed a deep and lasting relationship with Calia, Destron, and Sombren who I rescued from certain death at the hands of cultists? Me neither.

So rewind to the Longhouse Emperors twenty years before ESO. They make a deal with Mehrunes Dagon and specifically his dremora Xynaa to put a portion of his power into babies, creating the Ambitions. They plan to let this power mature in safe vaults and in the process mix Nirn and the Deadlands. When they sacrifice these Ambitions, they get the power and Dagon gets a permanent connection to Nirn and the Deadlands, allowing him to take over and making the Longhouse Emperors in charge of all Tamriel.

Enter Duke Varen Aquilarios. Exit Longhouse Emperor Leovic, pursued by the Duke.

Leovic dies, and now the Ambitions are up for grabs. Whoever kills them gets the power. A Dremora Xynaa trains one Ambition Sombren to control his portion of Dagon's power. Vandacia, Dagon's High Priest plots to find and kill the Ambitions himself.

As the Ambitions mature, the plot starts moving. Vandacia eliminates rival councilors of Leovic's, moving to find where the Ambitions are while framing the Dark Brotherhood for the murders. When assassins don't suffice, he attacks the remaining councilors in Leyawiin and is chased off by the Vestige.

Tracking down leads, the Vestige finds the Twin Ambitions in a doomvault. Destron and Calia are people, which is a great surprise to the Vestige, who was expecting Weapons. They bring the twins to Gideon because Leyawiin isn't safe.

The twins sit there like bumps on a log while the Vestige investigates yet more leads. Vandacia is throwing a party in the Deadlands where he plans to kill a captured Ambition for the power it'll give him. The Vestige crashes the party and rescues Sombren.

Sombren had training to use his power, so he tries to train the Twins. It all goes wrong, and we learn that their Dagon power gets out of control without training. That's very destructive, so we take them to meet Sombren's mentor, Xynaa, to get training.

You can guess where this is going. After a vision of a future where Vandacia brags that he sacrifices the Twins for power and gets to be Emperor of all Tamriel while Dagon rules over all of Nirn, Xynaa betrays Sombren and prepares to sacrifice the twins to Dagon so he can get their power and unlock the Gates between Nirn and the Deadlands.

We're the Vestige, so of course we rescue them. Vandacia decides he will get the Ambitions by hook or by crook and comes at us with an army. We defend the twins at Fort Redmane. Destron dies. Calia and Sombren help us defeat Vandacia and then skedaddle off to the Deadlands so Calia can learn to use her powers without destroying herself or other people.

So what did you stop?
Well, you stopped a bad future where Vandacia is Emperor and Mehrunes Dagon took over all of Nirn.

It's not over, since ZOS still has Deadlands. It's not really over until you've done Blackwood, Deadlands, and the epilogue you unlock by doing both.

Still, that's the plot of Blackwood. If it doesn't seem familiar, it's probably because most of the backstory is spelled out only in a Lorebook (Xynaa's Book of Contracts). Or at some point, you checked out and rode the railroad. I can't say I blame you.