What is the purpose of satire in "A Modest Proposal”?

The purpose of Swift's "Modest Proposal" is, first, to expose people's attitudes (particularly those in power) toward the poverty in Ireland and, second, for the author to give his own opinion on the situation.
Swift does this by combining realism with irony, absurdity, and parody. Swift writes the essay in the authoritative tone of an educated person using a structured, seemingly reasonable argument.

I shall now therefore humbly, propose my own thought, which I hope will not be liable to the least objection.

He also backs it up with the use of statistics.

The number of souls in this kingdom being reckoned one million and a half. Of these I calculate there may be about two thousand couples whose wives are breeders, from which number I subtract thirty thousand couples . . .

Most readers are taken in by his argument—up to the point where he starts talking about selling babies for meat. The essay then becomes an exercise in absurdity, but by then, Swift has already made one of his strongest points: just how easily a person in authority can fool the public into participating in something that is actually quite horrific. Yes, at some point the public will realize the authorities have duped them, but by then, it may be too late.

The purpose of the satire in Swift's "A Modest Proposal" is to draw attention to the plight of the poor Irish in 1729 and, by doing so, hopefully improve their lot and change the way the English treat them.  The narrator suggests a plan for the impoverished Irish to sell their one-year-old babies to the wealthy English as a new food source; this would give them a renewable source of income and reduce the number of beggars on the streets. 
To be clear, Swift does not actually want anyone to sell or eat babies.  He does not agree with his narrator. In Swift's time, the English had bought up most of the land in Ireland and had raised rents on their tenant farmers to the point that, although they could afford to pay the rent, they could not purchase any food or means of heat.  They had to beg for their subsistence.  Through his repugnant narrator, Swift suggests that, since the English are already figuratively devouring the Irish—they get fatter, it seems, while the Irish get leaner and leaner—it is not that much of a leap to suggest that the English actually devour the Irish.