It's specific to CPS form: every basic statement is a full function call, which moves to the next statement by calling another function (i.e. the continuation). Normally this would just be a very quick way to get yourself a stack overflow, so typical CPS-form compilers optimise (or require) the tail-call case, where you can pop the stack frame for the last instruction before moving on to the next. But for AD that's not a big deal, you just re-use those values in the backwards pass.
I doubt it's better than just allocating your own stack on the heap, especially when you have something like a differential equation solver with millions of statements, but it's still a neat trick.
I agree that the performance benefits of such stack allocation (over heap allocation) aren't quite clear in practice.
I feel the bigger win of delimited cont./closure-based AD approaches is that they can model the control flow of reverse-mode AD without AD-specific code transformations. Delimited cont. is especially great at making things modular: each differentiable function performs primal computation, calls the callback with primal result, then performs adjoint computation.
Is it a direct result of the semantics of delimited continuations? (perhaps because closures are passed as parameters and don't capture variables)
Or is it enabled via staging or optimizations?