the lord of the rings - What exactly happened to Gandalf's spirit after he "died" fighting the Balrog?

In the Two Towers movie, when Aragorn and company meet the newly reincarnated Gandalf the White, he has the following conversation with them

GANDALF VOICEOVER: From the lowest dungeon to the highest peak...... I fought with the Balrog of Morgoth. Until at last I threw down my enemy......and smote his ruin upon the mountainside. Darkness took me......and I strayed out of thought and time.Stars wheeled overhead......and every day was as long as a life age of the Earth. But it was not the end. I felt life in me again. I've been sent back...... until my task is done.

Why is Gandalf speaking like he is one of the mortal men? As a Maia, Gandalf's spirit or ëala is his primary form and therefore, after the destruction of his physical body, the ëala should have survived on and returned to Aman from where he originally came. We see it described this way in the cases of both Sauron and Saruman except that they don't return to Aman. That is, when their bodies die, their spirits rise out of the mortal remains and are "blown" away.

To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing. -Scouring of the Shire, RoTK

But Gandalf here doesn't remember the details of what happened after he died because he has just been reincarnated again. But that goes against much of what Tolkien thought about the Maiar. For a Maia, their spirit or ëala exists independently within the world and the physical forms they assume are just like clothes put on and cast off at will. But here, why is Gandalf talking like he was expecting his physical death to be the permanent end - "It was not the end. I felt life in me again"?

Answer

Wizards are Different

The Wizards aren't entirely like other Maiar. While you're generally correct that the Valar and Maiar aren't tied to a specific physical body, and therefore suffer the destruction of their physical forms the way we Incarnates suffer a torn shirt¹, the Istari are called out as having been put into "real" bodies; Gandalf isn't merely a Maiar spirit who's robed himself in a physical form, he's been incarnated:

For with the consent of Eru they [the Valar] sent members of their own high order, but clad in bodies of as of Men, real and not feigned, but subject to the fears and pains and weariness of earth, able to hunger and thirst and be slain; though because of their noble spirits they did not die, and aged only by the cares and labours of many long years.

^{Unfinished Tales Part 4 Chapter II: "The Istari"}

By 'incarnate' I mean [the Istari] were embodied in physical bodies capable of pain, and weariness, and of afflicting the spirit with physical fear, and of being 'killed', though supported by the angelic spirit they might endure long, and only show slowly the wearing of care and labour.

^{The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ (Draft). 4 November 1954}

I tend to look at the matter this way: there is a being called Gandalf, and a being called Olórin. While related, the two are not quite identical; among other differences, Gandalf is capable of being killed in a way Olórin isn't. Even though the death of Gandalf doesn't destroy Olórin, Olórin can't simply bring Gandalf back to life; that's against The Rules.

So what happened to Gandalf when he died?

We don't know; Gandalf himself is unwilling to discuss it further (though I would dispute your claim that he doesn't remember; he's quite cagey on the subject):

Then darkness took me; and I strayed out of thought and time, and I wandered far on roads that I will not tell.

^{The Two Towers Book III Chapter 5: "The White Rider"}

Some believe that Olórin left Eä altogether, and went to the Timeless Halls, though I disagree with that reading. Another possibility, and the one I favour, is that he returned to Valinor.

Unfortunately, Tolkien himself has no further insight to offer.

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¹ There are other complications to this highly generalized statement, but I'm not going into them here.