In a nutshell, the problem in Dublin circa 1700 (and the problem Jonathan Swift wrote about in A Modest Proposal) was extreme poverty. The narrator of the compact Proposal specifically highlights the plight of poor families with many children, and also the generally low rate of employment in Ireland at the time of the essay's composition. This poverty led to many social problems, and the narrator points out that lack of gainful employment has forced many Irish citizens into the army or slavery. As one might expect, Ireland's poverty is also accompanied by extreme hunger.
Swift's satirical proposal (the cannibal consumption of Ireland's young) is meant to combat both this extreme poverty and hunger, as eating children will simultaneously reduce the cost of raising a family while also providing a reliable food source. This suggestion is, of course, appalling, but it's also brilliant as a literary device, as it symbolically represents the fact that Great Britain, a primary cause influencing Ireland's poverty, was figuratively devouring Ireland's impoverished population.