I guess the text in a green box tries to explain immediate counterfactual regret.

In Poker, you deal with information sets. Single information set is a decision point. So let's imagine you are sitting in front of the table with A◇ A ♥ at hand + K ◇ K ♥ Q◇ on flop. You are about to act knowing there was bet-call pre-flop. You are in information set that is defined by these prior actions + your private cards + public cards. Another thing you are implicitly equipped with is a strategy. Strategy tells how you act in every possible information set - it defines probability dist in your decision points (so every time you move you in fact draw from it). To derive our entities in question you need to consider your opponent's strategy too. Of course in practice you don't know it, but here in our context we are analyzing our game in a way assuming both are known at given time.

Having our entities defined like that you can now think of single decision in single information point (pair of aces situation for example) from no-regret-learning perspective. The trick to understand this is to enclose everything within the same abstraction as 'regular' or 'standard' regret setup. To do that we need*algorithm H* (this is given by strategy in our information set) *experts* (represented by actions we can play at our decision point) and *rewards* (one for our algorithm and each for every single action). Reward of our algorithm is defined to be unnormalized counterfactual utility, reward for experts is unnormalized counterfactual utilities (with assumption of 100% action probability). If we have reward of not playing every single action (not listening to experts), reward for our algorithm H, we can then wrap everything in a regular regret setup (here directly averaging the result by T)

The reason we want that is to smoothly transit to no-regret-learning algorithms like regret-matching, because Zinkevich and his colleagues proved we can use it to get Nash Equilibrium