Concerning Swift's "A Modest Proposal," Swift had previously tried conventional writing methods in attempts to convince the wealthy English, the wealthy Irish, and the English government that oppression of the Irish poor needed to be stopped. Near the end of this ironic essay, Swift actually lists the solutions he had previously suggested: they're in italics, to separate them from his ironical proposal and to let the reader know that these are, indeed, serious proposals. He lists, for instance, the idea that landlords "have at least one degree of mercy toward their tenants." The italics are Swift's.
Since his previous efforts showed little results, he wrote "A Modest Proposal" in an attempt to shock his targets into doing something positive to help the Irish poor. The verbal irony allows him to do this. He couldn't seriously propose something like this, people would have just dismissed him as a lunatic, and, since he was on the Irish side of the situation, he couldn't honestly have done so, he wouldn't have wanted to. So, since it would make no sense to do it seriously, he does it ironically.
The essay is shocking, but once the reader "gets" the irony, it's also hilarious and entertaining.
Another benefit of verbal irony is that the reader experiences a feeling of discovery and even superiority. The reader discovers the true meaning behind the essay, and feels superior because he knows something the "targets" apparently do not.
In the end, however, the last bitter irony was at Swift's expense. Though the essay is still read and admired today, it accomplished nothing to help the Irish poor.
In general people use irony to emphasize a point. When we use words in ways where we mean the opposite of the words' literal meanings, it for some reason heightens the effect that the words have.
This is why Swift uses verbal irony in this essay. He is trying to really drive home to us how bad conditions are in Ireland and, more importantly, how poorly the government has done in trying to fix those conditions. By using verbal irony, he makes his points more vividly than he would by simply using words whose literal meanings coincide with what he's trying to get across to the reader.