Do you think the skill of writing improves through reading books? And why?

Reading and writing skills are very much linked.  Once this is understood by the learner, reading and writing become enjoyable learning skills and with time each skill will improve.  Visual learning of new words and how to compile new vocabulary, helps a writer to engage in using new words in their own work.  Different styles of reading material goes hand in hand with different styles of writing styles.  It is important to read not only books but other sources e.g. newspaper articles and high quality magazines of interest. 

Yes, definitely!  Through reading, you pick up a lot of ideas to write about or gather additional references on the materials you are writing.  You will also get significant insights from the writer of the materials you read.

Your writing skill improves with reading books and other knowledge products. It increases your vocabulary, provides you new information on use of the syntax, grammar, words and behaviour of characters. Reading and writing are inseparable. It is essential to be a good reader to become a good writer.

The basics of writing can be learned through reading the good, mediocre as well as the "bad" out there. By exposing yourself to styles, syntax structures and vocabulary, you will be able to come up with your own style and learn ways to express yourself especially if you are not a native speaker. 

I always advise my students to read in order to learn English.