I believe the popularity of Catan lies in the fact that it is extremely approachable while forcing players to make strategic decisions. The approachability comes directly from the small number of components and actions that the player must consider while playing. The strategy lies in determining which components to focus on first, and which can be obtained later. While the breadth of strategy is not large, it is large enough to be interesting, but small enough not to be overwhelming.
I'm going to approach the deconstruction by first recognizing the predominant strategies, and then discussing the game from the beginning.
Although I have not read extensively on strategies, it seems that two strategies are recognized the most. First, there is the strategy of focusing on brick and wood. I call this the "expand early" strategy. This allows for quick expansion with roads and settlements, and the possibility of obtaining the longest road victory points.
Second is the strategy of concentrating on ore and wheat. I call this the "city building" strategy. This allows the player to upgrade his settlements quickly and to utilize the development cards with the possibility of obtaining the largest army victory points.
I would contest that I have seen a third strategy work quite well: make sure that you are producing at least one of everything. Personally, I usually find it difficult to find locations that have good probabilities that all cover all five resources, but I have seen this work for some people.
Perhaps the single aspect that makes this game incredibly replayable is the board design. By randomizing the placement of the resource hexes, players have to adapt their strategy with each new game. Players cannot develop one tried and true path to victory -- what may have been a successful strategy in one game may fail in the next game.
However, the order in which the number tokens are laid out remains the same every game. The start point of the tokens and the location of the desert hex add a little variation to the final layout of the tokens. After several plays, you do start to see similar patterns emerge. In the end, the set up of the tokens was carefully designed so that two tokens with the same number would be next to each other. Additionally, no 6 and 8 (the highest probability production rolls) would be next to each other. While it may seem minor, this allows the board to be random while maintaining a certain amount of balance.
Determining turn order always seems to be a bit of a bear to me. Highest roll works fine, but I always feel that the second and third players (whether it's a three-player or four-player game) have a slight advantage. Perhaps it's because by the time it is time to place the fourth settlement, all the best production spots are taking. Additionally, by the time the first player is able to place his last settlement, there are very few spaces worth taking. This being said, it is probably only in my head that it seems that the second or third player wins every game I've played.
Players start with two settlements for two very important reasons. First, (despite what I wrote above) this gives a certain amount of equality to the player's opportunities when choosing their starting positions. By "snaking" the order of placement, the start positions are probably balanced the best way that they can be.
Second, by starting with two settlements, the players have the option of pursuing either of the two dominant strategies. With less than two settlements, the city-building strategy would fall flat if the player could not expand. With more than two settlements, the expand-early strategy would not be as powerful as a city-building strategy as players can only build 5 settlements in total before they have to upgrade to a city anyway.
Finally, the players start with one resource for each hex that is adjacent to their second settlement. While this may add a little strategy to the placement of the second settlement, it mostly serves as a way to jumpstart the game. Without this rule, several turns will be necessary before anyone will be able to build anything.
Players are limited to three main actions during their turn: produce (roll the dice), trade resources, and build. They can also play a development card if they have one.
Production is probably the most controversial part of the design. Many hold the belief that too many random results can ruin a design. I personally believe that the dice roll is random enough to be interesting because the results are not guaranteed, yet predictable enough that your settlement placement does matter. Additionally, by indicating the probabilities on the number tokens themselves with both pips and by the font size of the number, it is easy for players to immediately recognize where the higher probability numbers are even if they do not fully understand why they are the higher probability. (Anecdotally, one new player wasn't quite understanding the probabilities until he realized, "It's just like craps!")
Trading is the heart and soul of this game. The mechanic itself takes little design thought, but without trading this game would hinge more on luck than it already does. That said, the addition of ports plays two important roles. First, it solves any problem of stingy opponents that will never trade (Monopoly has this problem). Second, it increases the value of some of the coastal locations that only have one or two hexes adjacent to them. Successful strategies rely on the fact that you can trade with your opponents or the bank.
Building serves two purposes. The components built are either means to victory points (roads and development cards) or worth victory points. They are also the means to producing more resources. The problem this introduces is the "runaway leader" syndrom because the leader is usually producing more goods than the other players. This is mitigated by the hand-size penalty and the robber, which I will discuss later.
Finally, development cards play an interesting role in the game. From personal experience, they are not perceived by new players to be as powerful as they really are. They provide the compliment to a city-building strategy in the same way that roads compliment an expand-early strategy. Not only can you earn the Largest Army victory points (whereas the expand-early strategy lends itself to earning the Longest Road), but they provide an alternate way of gaining the resources you lack (whereas an expand-early strategy will allow you to expand to get these resources). In particular, the Monopoly card is perhaps a bit overpowered, especially if there is one particular resource that every seems to have all the time. For the most part they play an important role of making the city-building strategy viable (as well as giving you another way eto spend those pesky sheep!).
The gameplay is that simple. Once again we can see how the game is accessible to new players by limiting the number of actions a player can perform on their turn. Seldom does a new player find themselves saying "I didn't know I could do that!"
The hand-limit rule is quite important to the play of the game. First, it encourages players to build when they can, which in turn prevents players from hording cards in order to gain many victory points on their turn. It's not that this doesn't happen, but the hand-limit makes it a riskier strategy.
Second, it hurts players that are producing a lot of resources more than it does players that aren't. In this way, it helps the players that are further behind more than it does the leading players. However, it does hinge on the roll of a 7, which is only a 1/6 chance of occurring. By itself, the hand-limit rule does little to prevent a runaway leader.
The robber introduces perhaps the best means of containing a runaway leader. When one particular player emerges as the clear leader, it is common that all the other opponents will pick one of the leader's best production spaces to lay the robber on. However, the "best" space is purely subjective, so it isn't always a clear-cut decision. Perhaps the best effect of the robber is the lobbying that occurs from all the players as to which hex should receive the robber. However, the robber can be frustrating, especially if played early on a hex that is important to a particular player, and it doesn't move for several turns. For this reason there is a common variant or house-rule that allows 7's to be re-rolled for the first few turns.
The robber also provides a medium for a savvy player to acquire the resources he needs through deliberate use of the knight/soldier cards. If a player watches what resources other players are acquiring and spending, he can play a well-timed robber on a player that holds a key resource that he needs. Although this may not guarantee he will get that resource, if played at the right time, he can increase his odds of getting the resource he wants.
The game ends at 10 victory points. It is difficult for me to discern whether this was a "nice round number," or if it was determined in playtesting that it felt like a good number to end on. My initial reaction would be to guess it is the latter, as the game does not seem to end too early for players that may have a slow start, but it also does not drag on while everyone is waiting for someone to get that last point.
My main complaint about Settlers of Catan (probably even moreso than the luck of the dice) is that the game can easily take two hours to play. For this reason, a common variant is the "quick start" which has the players add a city to their second start location instead of a settlement. This does benefit the 3rd or 4th player the most as their 1st and 2nd locations are basically interchangeable, but it probably doesn't benefit anyone so greatly as to upset the balance of the start positions. Several official variants and expansions actually use this starting mechanism in the rules, but it could be for reasons other than shortening the game.
Settlers of Catan is a beautifully designed game. The simplicity of the game allows new players to pick it up quickly. Even with the low learning curve, players will want to come back to perfect their strategy. The random board layout creates a new challenge that forces players to adjust their strategy without forcing them to abandon basic strategies.
Once the players are comfortable (or even bored) with Catan, the base game lends itself to expansions and variants. With 3 official expansions and several unofficial variants, it is easy to see why Catan is in many gamers' libraries.