I compete in the expert division of Nat'l Scrabble Assoc. tournaments. My interest in Scrabble is mainly due to the unusual number of ways it challenges you. Obviously there's a brute analytical side to it, similar to chess (although chess is overwhelmingly analytical, and little else), especially in the Scrabble endgame. If you have tracked tiles accurately, you know what your opponent has left, they know what you have left (if they've tracked also), so now you're predicting the opponent's possible responses to each of your possible choices, and how well each scenario will play out.
But until the endgame, you're relying on a sense of probabilities as to what you will draw, what the opponent might have, or how much danger are you actually inviting if your play is vulnerable to the X, for example.
There's also an artistic side to it: conceiving possibilities, having good enough board vision to find great plays that hook in unusual ways. There is a psychological aspect to it, more akin to poker, that comes into play when you're considering trying to play a phony (or possible phony) and trying to judge ahead of time whether your opponent will accept it or challenge it.
Then there's also brute force memorization. Remembering RAINBAND is a word, and that it can be front hooked as TRAINBAND, even though you may play 100,000 games and never use this knowledge.
There's something inherently magical about marrying math and language. Scrabble is a math game involving words.
Math is logical, predictable, orderly. The layout of the board is mathematical. Language, especially the English language, is absolutely chaotic. Some words take S, some don't. Most Q words need a U, some don't. No rules, a few patterns, but always exceptions.
I also like the sense that every Scrabble game will be a different game, and that you have no idea of what sort of racks you are going to see or what words you'll play until it happens. In chess, of course, you start out with the same pieces in the same layout every time (and may have lots of games where the first few moves are the same).