wow. You know it's crap storytelling when even HaloStoryGaf doesn't even get it.
I would call this the worst finale of any Halo game but it doesn't beat the abrupt disappointment of Halo 2.
When Cortana is is inside the Forerunner computer mainframe, she split herself into several personality fragments. These fragments then converge on Didact on the hardlight bridge to pin him down to give the Chief enough time to disable the Didact and then set off the nuke.
The nuke doesn't go off instantly. There's a few seconds from the denotation to the actual explosion. Cortana finds an open link to the Domain, the Forerunner equivalent of the internet, except it has all their collective knowledge, history, culture and technology stored on it. By transferring herself to the Domain, she gains full access to the vault of Forerunner knowledge, including the ability to cure her rampancy. Which she does. She then uses the systems on-board Didact's ship to teleport the Chief to safety.
She realises that with the ship blowing up, she'll be trapped within the Domain, the only link to it was the ship and with the ship exploding, she'll be lost there. So she says goodbye to Chief, using the suit neural systems to talk with the Chief one last time. That's why she can touch and feel the Chief - she's projecting an image of herself directly into the Chief's mind.
She says goodbye because she doesn't think she can be restored from the Domain - when Forerunner die, their personalities and memories and knowledge were uploaded into the domain to preserve their life experiences so that future generations could learn from it. Cortana sees this process happening to her so she concludes that this is her death and it's final.
Chief then gets picked up, cue final cut scenes.
There's a massive backdoor here allowing Cortana to return from the Domain as she was never really alive in the physical sense to begin with, she's essentially a fancy AI or ancilla and as we have seen, they can be restored and transferred easily.
What is interesting is that when, not if, we meet Cortana again, we'll finally see a human AI having reached meta-stability. Unless something goes awry for her yet again, but I think they should really rest the Cortana is in trouble plot for any future Halo title. Two games is enough.