I once saw someone teach several new players 7 Wonders. After they had finished explaining what all of the cards do, one of the new players finally asked, "So, uh, how do we get these cards?" "Oh yeah, this is a drafting game, each turn you pick a card from your hand and pass the rest to the next player..."
It really bothers me when people are bad at explaining games, because it has a huge effect on how well new players understand the game, and consequently, whether they end up enjoying it or not. A game that a player otherwise might have loved can be completely ruined by a poor rules explanation.
In general, I think any good explanation should start with the overall big picture, then work your way down to increasingly specific details. So, like "each player has his own deck of cards; the objective of the game is to get the most VP; these green cards are worth VP; when you buy cards, they go into your deck; here's what happens on your turn; here's what all the different cards do". I feel like I've seen way too many people try to start with "here's what the cards do; oh yeah, I forgot to mention each player has his own deck; oh yeah, here's what you do on your turn..." (using Dominion as an example, but the general pattern applies to all games). The cards' effects don't have significance if I don't know the fundamental mechanics of the game, or the win condition, etc.