What is the sarcasm in A Modest Proposal?

Readers who don't catch the sarcasm in Swift's A Modest Proposal would think him a cruel and barbaric man, indeed. This type of sarcasm which is used to criticize others' ideas is actually called satire. Swift's target audience included the rich landowners in Ireland who seemed either ignorant to or insensitive to the poverty all around them. Instead of writing with a serious tone and carefully outlining all his main points, Swift uses satire as a different approach in hopes it would sensitize the wealthy to the plight of the poor.
Swift opens by seemingly aligning himself with the wealthy landowners. He notes the many offspring that the poor have and then mentions that they are always begging for money. He notes that all these children contribute to the "deplorable state of the kingdom" and that they should at least find a way to create some use for all these kids who are serving no other purpose.
If the reader pauses here, it is easy to imagine the wealthy nodding their collective heads, ready for a means of lessening the burden of all these poor beggars on their streets. What they likely didn't see at this point is that Swift has drawn them in with satire.
His solution is to have poverty-stricken wives serve as "breeders" and to collect these poor children as a food resource. After all, he notes,

I have been assured by a very knowing American of my acquaintance in London, that a young healthy child well nursed, is, at a year old, a most delicious nourishing and wholesome food, whether stewed, roasted, baked, or boiled; and I make no doubt that it will equally serve in a fricasie, or a ragoust.

Swift continues with this "plan," noting that skins can be used to make gloves, and he notes that older boys couldn't easily be used because their amount of physical activity would create tough meat.
Shocking, right? And that is just what Swift is hoping for. He hopes the wealthy will see that the poor are suffering and that to ignore them is cruel. Maybe not as cruel as consuming their children, but he creatively uses satire (or sarcasm) to capture their attention and bring light to a relevant issue in his society. Swift wasn't afraid of using the shock value of satire to find solutions for the famine and starvation that he saw plaguing Ireland.

Sarcasm is one device that is often employed by satirists in order to draw the attention of the flaws of those individuals who are being criticized. Sarcasm is often hurtful in its purpose, intended to wound its victim, and Swift employs sarcasm through his misguided narrator, who says, of the poor Irish babies as a food source,

I grant this food will be somewhat dear, and therefore very proper for landlords, who, as they have already devoured most of the parents, seem to have the best title to the children.

Swift does not mean that the rich English have literally devoured, or eaten up, their poverty-stricken Irish tenants, but Swift employs sarcasm here in order to suggest that the rich English might as well have done so. Their treatment of the poor Irish is no less brutal or cruel than it would be to go ahead and physically eat them up. If they are going to metaphorically devour them, then, he implies, why not go right ahead and literally do it?
Swift employs similar sarcasm when he has his narrator suggest that the skin from these babies "will make admirable gloves for ladies, and summer boots for fine gentlemen." Hopefully, such a suggestion would be considered abhorrent by his readers, but Swift seems to want to say, again, that the rich English might as well do this because they've so willingly let these children suffer and perish as a result of their own greed. Is there a significant difference between allowing them to starve to death and wearing their skin as accessories? Swift insinuates that there is not.

Sarcasm—or, as it is often called, verbal irony—is saying the opposite of what you think. When a person falls down while walking across the room, a sarcastic response is "Graceful!" or "Good job!"
In his essay, Swift creates a seemingly clueless narrator who expresses the opposite of what Swift believes. This narrator suggests that the problems of the poor in Ireland can be solved if the poor sell their one-year-old babies to the rich English landlords as food for their tables. The narrator suggests that babies, served hot on the edge of a knife, would be a sought-after gourmet delicacy.
While the narrator thinks his is a perfectly reasonable and "modest" idea, Swift found the tendency to treat the "poor" purely in economic terms appalling. He wanted to create a character and a proposal that would shock his audience fully. He hoped in this way to motivate people to come up with a humane solution to the problem of Irish poverty. He meant exactly the opposite of what he had his narrator propose.