The idea is precisely that players do not chose actions, but only chose one action at the time at every node at which they play, based on their beliefs about the way other players and themselves will play at future nodes in the game (where beliefs are conditional on the history that led to that node).
The interpretation is letting players choose full-fetched strategies is equivalent to letting players rely on a computer program to plays the game in their place. That is, they can commit via this computer program to playing a given action at each node.
Such games with commitment devices are in essence very different from games in which the actual players have to repeatedly chose an action at each of their decision nodes. When actual players play at nodes, players have to form beliefs about the way other players and themselves will play at future nodes, and these beliefs may depend on the history that led to future nodes.
For instance, in a Stackelberg game, the leader could believe that the follower will be rational (i.e., utility maximizing) if the leader plays "Low production", but will be irrational (i.e., non utility maximizing) if the leader plays "High production". Maybe the leader anticipates that the follower will be angry if the leader plays "High production", and that, blinded by her anger, the follower then then want to retaliate.
If the follower could have committed through a strategy, the game would have been completely different. Maybe the follower could have committed not to retaliate before she gets angry, and she cannot help her desire to retaliate anymore. But here the idea is that the actual follower has to choose an action later in the game given what the leader chose at the root node. Therefore, the behavioral rule through which the follower chooses an action at a node (e.g., utility max vs. non utility max), and the beliefs of the leader about these procedures may depend on the history that led to that node too.
This opens the way for many new outcomes of the game that would not have emerged from classical game theory. From a conceptual point of view, it also switches the focus from solution concepts to epistemic and behavioral assumption (i.e., from classical game theory to epistemic game theory). Instead of identifying a set of reasonable outcomes (e.g., Nash equilibrium outcomes) and look at the strategies that match these outcomes, one identifies reasonable properties of players' behaviors and beliefs (about each others' beliefs and behaviors), and derives the conclusions of these epistemic and behavioral assumptions for the outcome as the game unfolds.
Now, this is just to give some meat and intuition to Battigalli's framework, and it is does not do justice to the richness of the framework (in part because I don't know much about his work other that the video you linked to). If you haven't done it yet, I strongly recommend that you watch the whole video. I think Battigalli does a great job at making his framework accessible. He also present helpful and intuitive examples to connect his epistemic approach to "classical" game theory by identifying simple conditions on players beliefs and behaviors that allow to recover classical solutions to games such as backward induction.
