When someone is called the epitome of style, they are being compared, at the root, to a book summary.
Greek epitome meant "an abridgment," and also "a cutting on the surface." It comes from the verb epitemnein, "to cut short, abridge." That joins epi, "into," to temnein, "to cut." Etymonline traces the word through Latin epitome and French épitomé into English.
English first used it for exactly that kind of shortened text. Merriam-Webster gives 1520 as the first known use, meaning a summary of a written work. Etymonline describes the sense as a brief statement of the chief points of some writing.
Epitomes mattered more in the ancient world than the word now suggests. Sometimes only the short version survived. The Philippic Histories of the historian Pompeius Trogus ran to 44 books and is now lost. Readers know it mainly through Justin's Epitome and a set of brief anonymous summaries called the Prologi. Wikipedia notes that Justin often left out Trogus's geographic and ethnographic material. What readers have today is the cut-down copy.
The modern meaning grew out of the old one. A summary holds the essential points of something larger in a small space. By about 1600, Etymonline says, epitome could mean a person or thing that typifies something. Merriam-Webster's senses show the path. It lists a typical or ideal example first, then a summary, then a brief or miniature form of something.
The verb followed the same route. Epitomize meant to shorten or abridge in the 1590s. By the 1620s it also meant to typify or embody.
The cutting root has relatives. Etymonline lists tome, which began as Greek tomos, "a section, piece cut off," before it meant a volume. Atom is the same root with a negative in front: atomos, "uncut, indivisible." An epitome, a tome and an atom are all, in their Greek origins, ways of describing a cut.