书 (traditional 書) originally pictured a hand holding a brush (聿) over something — the act of writing. From 'writing' it came to mean the thing written: a book, a letter, a document. The simplified 书 is a fast, brush-like shorthand.
书 means both 'book' and 'to write/writing'. 读书 (read books) is the everyday phrase for 'to study' — for two millennia, education in China WAS reading the classics. 书法 = calligraphy ('the method of writing'). 图书馆 = library ('picture-book hall'). 看书 (with 看, which you know) = 'to read a book'.
A hand and brush, simplified to a few quick strokes — WRITING, and the BOOK it makes.
读书 ('read books') is how Chinese says 'to study' — because for 2,000 years, study WAS reading the classics. 看书 (with 看) is 'to read a book'.