Why did Alexander Pope write about criticism in verse?

Your question, I think, is really two questions in one. First, why was Pope interested in writing about criticism? And second, why did he choose a poetic format, rather than prose, to deal with this topic?
Pope, like most creative artists, was extremely sensitive about what people thought of his work. At the same time, he himself was critical of other writers. The Essay on Criticism was written in 1709, when Pope was only 21. Much later in his career he produced the mock-heroic satire The Dunciad, an attack on other writers, in particular the playwright Colley Cibber, whom Pope considered incompetent. But throughout his life both Pope and his close friend and fellow satirist Jonathan Swift self-consciously believed in, and expressed, their sense of being above the rest of the English literary world. This fact was noted by Samuel Johnson in his Lives of the English Poets, where he observed that in the correspondence between Pope and Swift the two expressed such a sense of superiority to other writers, and to others in the intellectual world generally, that a credulous person, reading their letters would think that the other writers of the early eighteenth century produced virtually nothing of value. In his Essay on Criticism, when his career was just getting started, Pope felt the need to express, in a didactic fashion, the principles of criticism that were generally accepted in his time. In some sense the young genius was establishing himself as the arbiter of taste in the literary world and also making the statement that critics themselves, to put it colloquially, often did not know what they were talking about.
The second part of your question involves the format in which Pope chose to make this statement: the standard poetic form of the age, rhymed iambic pentameter couplets, as Pope's idol Dryden had used earlier, and as Samuel Johnson and many others were to use later in the eighteenth century. Pope and others held the view that ideas are impressed more forcefully on the reader and become more memorable when they are expressed in meter and rhyme. Just as with his later Essay on Man, the ideas in the Essay on Criticism were not ones Pope himself originated. His intention in both cases was not to put forward new ideas, but to express, in as beautiful and memorable a way possible, ideas that were the standard beliefs of his time, as kind of celebration of neo-classical aesthetics and, in the Essay on Man, the philosophical beliefs of his time.