Rust futures are "just" structs with a poll() method. The poll() method is a function like any other, so it can have local variables on the stack as usual, but anything it wants to save between calls needs to be a field of the struct instead of a stack local. The magic of async/await is that the compiler figures out which of your async function's variables need to be fields on that struct, and it generated the struct and the poll method for you.
I have a blog series that goes into the concrete details if you like: https://jacko.io/async_intro.html
I see. The Rust implementation effectively splats out the transitive closure of all your callee stack frames upfront which would enable continuing previously executing async functions.