Swift, in the persona of a learned scientist, attempts to tackle the chronic problem of over-population in Ireland. In turn, this problem leads to lots of other problems, such as poverty, starvation, and an excess number of Roman Catholics.
At that time Ireland, an overwhelmingly Catholic country, was run by the Protestant British, who systematically repressed the majority religion. As part of his recommendation, Swift advocates the breeding of poor Irish children for meat as a way of reducing the number of Catholics in the country. The understanding is that this will make Ireland much easier for the British colonial authorities to administer.
In addition, Swift's modest proposal will also act as a much-needed stimulus to trade in such an economically backward country. As with all of the mock scientific paper's recommendations, this is highly satirical. Swift was a constant critic of the economic exploitation of colonial Ireland, which he saw as keeping the country from becoming prosperous. Instead of stimulating trade and investment as they were supposed to do, the colonial authorities were perfectly happy to let the Irish economy stagnate, the better to enrich the British economy.
One can imagine many administrators warming to the idea of breeding children for meat as a quick fix solution to Ireland's economic problems, and as a way of fulfilling a number of other key policy objectives. This is why Swift's satire is so remarkably effective. In their unguarded moments, hard-pressed colonial administrators may very well have entertained such an unspeakable solution to the many problems that they faced.