Books expand your mind and have the power to increase the expanse of your thinking exponentially. More significantly however, a good book has in it the power to transform you, the single, most important person who will bring about a positive change in your life. In the answer that follows, I unabashedly borrow and retrofit a couple of ideas I came across in one of Robert Kiyosaki’s books so that I can answer your question. I do not recall now which particular one they came from though.
Generally speaking, there are two things that you can take away from any book that you read, and these are: a) content; and b) context.
Content is what is usually taken to mean as knowledge and skills, the kind of specialized knowledge required of you to operate better in a particular area of your life. For instance, if you read a book on a hard skill like, say, computer programming, or math, or investing, or even a book that shares valuable soft skills like say, learning to interact with others more meaningfully, you are effectively expanding your content. Broadly speaking, such books can be classified as hows.
On the other hand, there are books that help you re-align your mental context, that is, change your internal attitudes, like the ones that train you how to think positively, or help you expand your mental arsenal for dealing with life; in other words, books that tend to enhance your attitudes — hopefully for the better — either towards life as a whole, or towards a particular area of your life like your career or your relationships. In that sense, rather than helping you deal with the mechanics of things, books of this kind help you resolve your whys.
I hope this distinction between content and context is clearer now.
Next, coming back to your question, which is how do people get rich by reading books: Know that books do not just equip you with the skills required to get rich by expanding your content (by furnishing specific knowledge required of you to acquire wealth), but also by enhancing the context within which your mind operates. Books of the latter kind help orient your mind in a more positive and fruitful direction.
Keep in mind the be-do-have formula.
A good collection of books will help you be (by changing your mental context); which then helps you do (take specific action that leads to positive results, and which requires specific content-based knowledge); with the final phase of this process being to have (where you enjoy the fruit of your efforts).
The whole point of knowing the be-do-have formula is that most people tend to get the entire process wrong, reversed in fact. They look at what the rich have, and they then go out and buy the same stuff on their credit cards so that they too can have, without realizing that to have, you must first do. But then to do, you must first be. I hope this is beginning to make sense.
Good books help you understand this process and right it, one area of your life at a time. Because a good collection of books will help you change your context to a more positive one, so that you are motivated to change yourself to be, so that you are able to do what you need to do to eventually have.
Image courtesy: By Charles Dana Gibson - Transferred from en.wikipedia to Commons. Original uploader was Hugh Manatee at en.wikipedia, Public Domain, https://commons.wikimedia.org/w/index.php?curid=6151973