First, we can never entirely be sure of the motivations of historical figures as we lack access to their inner thoughts. Also, even when people do claim a certain "true purpose", human motivations are often mixed, blending the conscious with the unconscious and the noble with the pragmatic. What we can determine is the main points he was arguing and how they fit within the system of beliefs and commitments we find in both the text itself and other elements of his writing and biography.
Swift was born in Dublin in 1667 to an English (Protestant) family, part of the English elite that ruled Ireland and controlled much of the wealth and land of the country as colonial conquerors ruling over a country that was Roman Catholic in religion and Celtic in language. Irish Catholics lived under many oppressive rules and were barred from formal education and holding most important civic positions. Many lived in extreme poverty.
Swift's own career included education at English Protestant schools in Ireland, a period employed by the retired English diplomat, Sir William Temple, and then a career combining writing, politics, and priesthood in the Church of England, culminating in a position as Dean of St. Patrick's Cathedral in Dublin beginning in 1713. Between this period and the composition of "A Modest Proposal" in 1729, Swift wrote several patriotic Irish works, urging better treatment of Ireland and the Protestant Irish clergy (who were part of the Church of England) by the English government. Given this personal history, and Swift's explicit concern in other works about the condition of the Irish poor, we can be relatively certain that the purpose of the satire was to draw attention to the plight of the Irish poor and to urge reform in the way that the Irish poor, especially women and children, were treated.