One thing I keep coming back to when it comes to rulebooks and rules explanations, is that players need to understand what they actually *do* in the game.
This is often conflated with which rules they have to follow, or how an action is translated into the game's theme.
But the first step is knowing you have to play cards, or keep others from placing tokens, or combining actions, etc. It's only after understanding these core elements, that you can expand into theme and the limits set by rules.
@Georgios Makes me think of the way I teach games to new players. I tend to start with the ultimate goal (having the most VP, for instance) then doing a quick overview of the gameplay in reverse order (starting from the end). It gives a sense of what will need to be done to accomplish this "ultimate goal". Only then will I start explaining what needs to be done (draw cards, play cards, place miniature, roll dice, etc.), since the goal and the theme are already covered in the overview.
@jonathang To be honest I believe the secret ingredient to every great rules explanation is a confident, unflappable explainer.
If players trust you to guide them through any difficult situation during the game, they're more likely to overlook or push aside whatever it is that they haven't understood right now.
That said, an organising paradigm as @TimClare mentioned is very helpful.