What's the point of stack judgement in CBPV? Call-by-push-value (CBPV) introduces two main families of types, values and computations, and their corresponding judgements. However, in some extensions/variants/adaptation of CBPV, there is a third judgement: stacks. What is the purpose of this judgement? Why it was not present in the original CBPV? What do we gain when we introduce them?

The kind of extensions I am talking about are for example Levy's CBPV with stacks, or Egger et al.'s Enriched Effect Calculus, or the calculus in Ahman et al.'s Dependent Types and Fibred Computational Effects.
The point of stacks is that they are in a sense the dual concept to computations.

A computation does not run in a vacuum. It is always "surrounded" by some sort of an environment, or evaluation context, telling us what the current "state of progress" is, or is recording "where we are, and where we're going". Often this sort of information has stack-like behavior, so we can call it simply a stack.

The duality between computations and stacks can be made quite precise and is useful in formalizing and discovering various mathematical aspects of computation. I am not sure where you're expecting a more technical answer, but that's the gist of it.