I have listened to a fair number of people explaining board game rules and I feel quite strongly that most of these explanations present too much information at once. Often someone reads the rules, and then basically tries to present them all at the same time, which leads to people unfamiliar with the game feeling overwhelmed and not processing the information. Later, when playing, you can observe the actual learning happen, once people ask what moves they can make and observe what the other players do.
I think that it is much more efficient to simply drop the first part and go straight into the game. Whenever an action is required, simply tell the person their options and focus on reducing the information as much as possible.
Like, if you were teaching someone to play chess, you could prepare the board and play as white, while your friend plays black. Then move a pawn while explaining the way pawns move. Let your friend move a pawn. Continue until the capturing rules for pawns become relevant. Explain those. Then, one-by-one, introduce the other pieces and their movement over multiple turns, when they can actually move. At some point, a check will happen (or you can make it happen). Now you can go into the check rules, and how a game is won. There is no need to explain everything in the first game either, things like castling, pawn promotions or en passant can be postponed until they become relevant.
Also, there is no need to involve strategy. The focus should be on teaching to play, i.e. take valid game actions, not on how to play well. In addition to being unnecessary information, there is also a lot of fun in figuring things out by yourself.
Usually, for board games the designers thought about the best way to teach the rules and the manual is helpfully structured in that way. So it will first describe how to prepare the game, then the turn structure, then the possible actions, and so on. Often you can start a game parallel to reading the rules, which is handy when no one knows the game.
Simplifying the game, as in the linked article, is also quite useful. There are some good examples of explicitly changing the games to be simpler, but you can also do some more subtle manipulations. For example, in Go (the board game, not the programming language) beginners play with the full rules (which are already simple enough) but on a smaller board, which leads to shorter, less complex games. In Magic: The Gathering, you can teach someone by preconstructing the deck to contain only simple cards. (I have seen some board games incorporate this as well, but there it is usually framed in terms of 'extensions' to the rules.)