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.
An important way in which writing makes an exact man is that a conscientious writer must always be searching his mind, as well as his dictionary and his thesaurus, for the exact word to express his meaning. Many famous writers had a very hard time writing. These include Ernest Hemingway and Honore de Balzac. Here are some pertinent quotes:
Composition is, for the most part, an effort of slow diligence and steady perseverance, to which the mind is dragged by necessity or resolution, and from which the attention is every moment starting to more delightful amusements.
A writer is somebody for whom writing is more difficult than it is for other people.
Writing is a dog’s life, but the only life worth living.
Flaubert, the great French realist novelist, best known for his novel Madame Bovary, made a lasting impression with many writers with his insistence on finding le mot juste, which means the right word. One word can sometimes do more than a hundred words. Sir Francis Bacon must have been guided by that principle because his essays are invariably short and even seem to compress or synthesize a great deal of learning and wisdom in a small space.
If a writer cannot come up with le mot juste immediately, he can just leave a blank space and come back to it later. Sometimes the right word is hard to find, even though we know it must exist, and must have existed for centuries. Often it will come to mind spontaneously out of the blue. It is satisfying to a careful to be able to put the right word in the right place.