It bugs me that I am right with Tite Kubo (author of Bleach) about the Ichigo's Final Getsuga Tenshou right up until the sword, Zangetsu, attempts to explain in the dreamworld what is happening. I feel like he's close, but doesn't quite reach, the perfect thematic explanation.
It's been said all throughout the series that the Zanpakutou is "part of the user's soul" as it were. Thus, the short short answer to the question is that the sword gives up its identity and returns control over the power to Ichigo. Ichigo, in his short-sighted mentality of conquering the challenge ahead of him, then uses up every scrap of power, burning out the source of his power entirely. It's like taking control of your adrenaline glands and forcing your entire body to completely wreck itself in a single, sustained burst of activity.
The longer answer is that the existence of the Zanpakutou was literally created; it is a tool created by ancient powerful spiritual whozits to give power to those spirits who will be protecting the spirit world and the real world. Part of the duty of a Zanpakutou is to maintain itself, because it is a spiritual "machine" which cannot depend on anyone else but itself for maintenance. Later, researchers etc could repair them, but almost no Shinigami/death gods need or use those facilities because they for the most part maintain themselves.
As part of maintaining themselves, these Zanpakutou must constantly hold back. They should not damage themselves or their user, they should listen and obey, they should maintain their own identity and by doing so offer perspective that the user would not otherwise have, etc. They need to constantly be generating or gathering energy, setting some aside for their own needs, and only letting energy out that they know how to control.
Ichigo is not a Zanpakutou and has a lot of conflicting influences that add up to his source of power. Indeed the idea of the "Final" Getsuga Tenshou is probably something that most Shinigami could not do. Canonically Ichigo has four power sources: Shinigami by birth (and as a gift from Rukia), Visored (Hollowification), Quincy by birth, and Fullbring (Mother attacked by a hollow during pregnancy). Each of these grants Ichigo a different way to use spiritual power to perform supernatural feats, although at the time Ichigo had only been trained in half of them. If Zangetsu--who more accurately is an amalgam of the power sources inside Ichigo--is willing to step aside and directly grant Ichigo access to all of the power that he naturally has, then that would account for the form and function of the "Final Getsuga".
It's worth taking a dalliance to talk about Ichigo's mannerisms in his fight against Aizen. If I could describe Ichigo in two words, I think they would be "Unimpressed" and "Frustrated." Ichigo has spent an enormous amount of time catching up to Aizen, and his final ace in the hole is literally to throw away the tool that was holding him back and take on full responsibility for using his powers himself--with the probable consequence of losing his abilities forever. The only way he has become strong enough to overcome Aizen is realize that the source of his strength, which is buried inside himself, is separate from him, and only by casting it away can he reach the height he needs to.
In contrast, Aizen basically since his dramatic reveal has been coasting on brains, his sword, a shit-ton of Reiatsu, and the giant pair of balls it takes to go one man against the entire world. He covets the Hougyoku--the spiritual transformation bauble--and expects it to allow him to transform into something truly transcendent. It is fair to say that he is expecting someone else to do the work of advancing him to the final stage of his powers.
This contrasts sharply with the Final Getsuga. If you understand it as Ichigo taking direct control over, and direct responsibility for, his power, then you could not have a more opposite character than Aizen screaming maniacally at the jewel in his chest, telling it to give him more power.
So when you look at Ichigo's eyes throughout the entire fight with Aizen, take that into consideration. After all of that training, and all of the sacrifice, Ichigo looks like he has to handle a spoiled, angsting teenager--someone truly beneath him not by birth, but in attitude. Word after condescending word pours out of Aizen's mouth, and Ichigo just sees him for what he is, and doesn't give a crap about anything he says. He knows first hand what it takes to actually get to Aizen's level by his own power, and it's clear that Aizen does not.
I love the concept but I can really imagine the author finding himself written into a corner and just not quite being able to find the words for it. Which, really, I sympathize with, I really do. I think if he had found a different way of phrasing it, he probably would have altered what came later as well, but I dunno. Writing is like that.