The purpose of all satire is to hold up for public inspection behaviors of which the writer disapproves. These can be seriously bad behaviors (Juvenalian satire) or ones that are just foolish (Horatian satire). When these behaviors are exposed to public scrutiny, the author hopes, those who engage in them will be pressured to mend their ways.
Since "A Modest Proposal" is one of the greatest satirical essays of all time, we can assume that its purpose is to try to change behaviors Swift doesn't like. In this case, the main target of his disapproval is the way that the British government is ruling Ireland, which was a British possession at the time.
I believe that Swift hoped that his satirical essay would get people to think about what he saw as the evils being visited upon Ireland by British rule. Once they did that, he would have hoped they would pressure the government to change its policies.
He was hoping the British public would question the way the British government was ruling Ireland