The quotation to which you refer comes from Francis Bacon's essay "Of Studies," and is part of a longer quote in which he says, "Reading makes a full man; Conference a ready man; and Writing an exact man." Bacon wrote fifty-eight essays over several years and published a complete edition in 1625, The Essayes or Counsels Civill & Moral of Francis Bacon. Bacon's purpose in writing essays--which discuss moral, religious, business, and even practical subjects like gardening--is to create a kind of road map for proper human behavior for a man in politics or business in the important spheres of life. His essay "Of Studies," for example, in which he discusses writing, discusses the importance of learning:
Studies serve for Delight, for Ornament, and for Ability. Their chiefe use for Delight is in privateness and retiring; for Ornament, is in discourse; & for Ability, is in the judgement and disposition of Business.
In other words, Delight in studies allows a man to be happy and useful during private time away from business or other duties--primarily because he is learning something that will improve him; Ornament, by which Bacon mean understanding the rhetorical arts like argument and persuasion, allow a man to speak effectively to others; and Ability gives the man enough practical experience to understand political and business matters so that he can successfully manage his political and economic affairs.
When Bacon says that "writing makes an exact man," he follows that immediately by the warning, "if a Man write little, he hath need of a good memory." Bacon, who wrote hundreds of pages, in a style that we now call the "plain style," understood that writing--and this is an aspect we recognize today--helps a person remember complex matters because writing tends to imprint on the mind what a person writes. More important, however, is that Bacon was aware that writing, because writing must be precise to be understood, also forces the writer to think clearly about the subject. An axiom (a universally understood truth) of writing, encapsulated in Bacon's comment about writing and exactness, is that if a person cannot write clearly about a subject, he cannot think clearly about that subject--and that is why Bacon links writing with being exact or precise.