What problem does Swift identify in "A Modest Proposal"? What general solution does he recommend?

Jonathan Swift, himself originally Irish, was Dean of St. Patrick’s Cathedral in Dublin and thus a member of the Church of England as well as a distinguished writer.

Ireland at this time was primarily Gaelic-speaking and Roman Catholic. The English had fully conquered Ireland during the Tudor period and were English-speaking Protestants. Outside the Dublin "Pale", many of the Irish resented their English overlords and civil unrest was a perennial issue, with the Irish often aiding English enemies or rebels (such as the Jacobite uprising). In response, England's rule of Ireland was quite oppressive, restricting educational and economic opportunities for Irish Catholics. Partly in consequence of this, the Irish tended to be impoverished, ignorant, and at risk of starvation due to periodic famines. 

The immediate problem that Swift is addressing in "A Modest Proposal" is famine. He wrote during a period in which there were a series of bad harvests and many of the Irish were starving. He is particularly concerned with seeing mothers and young babies starving on the streets. 

In this satire, Swift proposes that much of this starvation could be averted if Irish babies were used or sold for food. He does not mean for this suggestion to be taken literally. Instead, it is intended to make the reader understand the dire nature of the Irish situation and make the English reverse some of the economic laws that had destroyed the Irish economy.