In Altered carbon, the human bodies are referred to as "Sleeves" and the process of transferring one's consciousness to another body is called "re-sleeving" (as you clearly mentioned).
According to the Alter Carbon wikia site (here):
The mind is no longer fixed to the body, because the stack contains the human mind. The mind (the DHF) can be transferred via “needlecasting” to any other stack in any other sleeve, provided that the stack is wiped before the transfer.
The sleeve you are born into is called your “birth sleeve.”
When a sleeve dies, but the stack is still intact, this is called “sleeve death.” It is not fatal to the mind, only the body.
So, the stack simply doesn't record human actions, IT LITERALLY RECORDS EVERYTHING. Then the re-sleeving is done by first wiping the sleeve's stack and then transferring the consciousness from the person's stack into the sleeve's stack.
But on the downside,
The human mind is biologically evolved to stay within the body into which it was born. If a person is resleeved too many times, the mind will rebel, and the personality starts to fragment. In short, resleeve into too many different bodies, too many times, and you go insane.
If a person resleeves into their own biological body over and over, however, there is no personality frag.
This is why Meths grow clones to resleeve into; they can live forever without any personality degradation or fragmentation.
And finally about "what is left for the higher functions of the sleeve's brain to do?". According to the book:
Cortical stacks serve as a receptacle for the human consciousness or Digital Human Freight (DHF).
In simple terms, they are like recorders; they take note of everything that happens. But still the brain of the sleeve is responsible for the sleeve's functionality. Take for example the action of thinking. The brain thinks, but the sleeve 'records' what the brain thinks. And all higher order functionalities are done by the brain. This is why rich people (who cannot afford clone re-sleeving) buy the sleeves that have great skills (like the one Bancroft acquired for Kovachs).