Among many definitions of "satire" the following is the most compelling regarding "A Modest Proposal":
the use of humor, irony, exaggeration, or ridicule to expose and criticize people's stupidity or vices, particularly in the context of contemporary politics and other topical issues.
Hence, this is exactly what Swift does. He exposes and criticizes contemporary politics and the way things are carried out in his society. He particularly attacks the cruel abuse and discrimination against Irish catholics, who are poor, disenfranchised, and still mistreated by English protestants.
Jonathan Swift takes on the voice of an extremely prejudiced English Protestant, and uses the argument of the Irish eating their own children (in order to decrease the amount of Irish Catholics) as an extreme way to exaggerate the usual anti-Irish discourse. This is his way of mocking the people who agree with it:
“For first, as I have already observed, it would greatly lessen the number of papists, with whom we are yearly overrun, being the principal breeders of the nation as well as our dangerous enemies”
Swift also creates satire by using a medium that is commonly used for serious purposes. Think of an open letter, or rather an editorial in an important publication, or even a thesis. These are often conduits of serious topics of discussion, where educated scholars propose meaningful solutions to current issues.
Just imagine taking that same, erudite tone of voice to propose that a group that disturbs the peace of a bunch of abusive, dominant overlords is encouraged to eat their young for purposes of population control. The very idea is unthinkable, and the exaggerated nature of the treatise itself tells that it intends to make mockery of the anti-Irish protestants.