Learning a language is a lot like learning to play a complex game. You can spend all your time learning the rules (grammar) and still not know how to play (have a conversation). Luckily, I have a bunch of techniques I use with languages that I can apply here, too.
The most important one is LIMIT. If I teach you German, I will start with only the present tense, with only a few objects (all of the same gender), and a few verbs. Yes, I realize this tiny subset of German is not nearly as interesting, beautiful, or powerful as you might find in poetry or speeches. That's OK. This is just how we get started.
With a game we can do something similar. We can pick a tiny subset of the game, teach it, play it, then add another BITE-SIZED PIECE, and repeat. Luckily Settlers is not nearly as complex as German, so it won't take long.
Another technique is PLAY NICE. People often feel vulnerable when learning something new, especially if they feel pressured to perform. Take gentle care of them.
Finally, make sure everyone quits while they're still having fun ("FULL"). Like eating, learning is only fun before you get full; if you keep going too long you'll vomit. Yuck.
I'm going to make this more concrete, but I'm going to do that by describing the rules that I'm cutting out. My list will only make sense if you already know how to play the game. If you don't, then skim for now, until you see the minimal ruleset below.
You can remove rules in the order listed; add them back in reverse.
8. No development cards. There's nothing wrong with development cards, but they aren't required for the mechanics of the game to play out. Set them aside for now. Also skip the Largest Army bonus. Since this reduces the possibilities for getting points, be sure to reduce the number of points required to win.
7. No robber. Without development cards, there are no Knight cards, so nothing depends on the robber. Robber is also a bit cruel. Out he goes.
6. No lost cards when rolling 7. This rule quite harsh, especially for new players who aren't sure how to use their cards.
5. No player trading. Trading is clearly an important part of the joy of the original game, but it's not necessary for the game to function. (Remember that we're not aiming for "great game" but for "easy to learn".)
4. No harbors. Flip the ocean pieces over so the harbors are hidden.
Do allow 4:1 trades with the bank, as this is required for the game to function properly. Without the risk of losing cards on a 7, players can hold as many cards as they need to trade for what they want.
3. No secrets. Since you can't steal resources from each other, and you're not trading, you don't need to keep your cards hidden. Everyone places their resource cards on the table face-up, so everyone can see. Then you can coach people on how to use their resources.
2. No roads. Now we're really getting lean! Players can place settlements on any intersection that isn't adjacent to another settlement. Remove the Longest Road while you're at it, and adjust the winning score. (Again, remember that the game doesn't have to be well-balanced, just easy to learn.)
1. Don't hand out resources when players place their starting settlements. It's just simpler.
So now the game looks like this:
Set up the board with the usual tiles, but with the oceans upside-down. Place number chits as usual. Give players the Building Costs cards as usual, but tell them to ignore the Development Card and Road entries.
Players take turns placing starting settlements, as usual. Don't give much attention to getting a great starting position; this game will be over soon.
A player starts a turn by rolling the dice. Distribute resource cards as usual. Player can build settlements anywhere, or upgrade to cities. Player can trade 4:1 with the bank as needed.Not only do we reduce the winning score because there are fewer ways to get points, but we should reduce it further to get through the game quickly. In my experiments today, 5 points worked well.
You can probably only play this way once, unless you're playing with 4-year-olds. So it's time to add some more rules. It is possible to add all the remaining rules at once -- that's still an improvement over the usual way -- smaller steps are probably better. You'll have to decide based on your situation. How eager are the players to learn more? Is there a lot of background noise that will tire players out? Will young children get tired of waiting to play while you list rules? Is your voice tired?
If the step is too small, players will get bored with the overly-simple game, and annoyed at the work of setup/cleanup that each game requires. Much of the joy of games is learning how to play them. Give players enough challenge so they get to experience that joy.
If the step is too big, people will get bored listening to rules, or overwhelmed trying to follow them.
Keep the winning score low for now. A small step of new rules combined with a high winning score means people get to know the game really well fast, and then get bored waiting for the game to finish.
It may not be a good idea to ask a new player to play the full game today. Help them quit while they are still having fun, and they'll ask to come back and play again another day.