In this essay Swift uses verbal irony to get his point across. Verbal irony is a disparity between what is said or written and what is really meant - we use verbal irony all the time in our lives when we comment upon things, for example, "I can't wait to get back home so I can start on my homework", whereas, obviously that is anything but the truth.
To convey verbal irony when we speak we can rely on our tone of voice to alert our listeners to the verbal irony in our speech. Writers cannot depend on tone of voice, so include so many examples of verbal irony that the reader cannot miss the point.
This essay is a classic example of verbal irony stretched to its very limit, from the title, "A Modest Proposal", which is anything but modest, to its ridiculous suggestion of eating Irish babes and comments on the relationships between the Irish and their English overlords. This excerpt is one of my particular favourites:
I grant that this food will be somewhat dear, and therefroe very proper for landlords, who, as they have already devoured most of the parents, seem to have the best title for the children.
Here, the humour is based on the multiple meanings of "devoured", which in one sense refers to how the Irish adults have been made poor by rents, but another sense refers to a metaphorical devouring that clearly establishes Swift's opinion of how the English are acting in Ireland. Eating infants, therefore, is the only logical conclusion to such a policy.
In addition, the irony is increased by a constant reasonable tone of modesty, combined with the assumption of a voice of a practical economic planner. The speaker pretends to be full of common sense and completely objective, and at times, even sensitive and kind. This disparity between tone and content gives the essay a real bite.