The answer to this question can be found in the very first line of the essay. What the author finds melancholy (this word means sad or lamentable) is the extreme poverty of the people of Ireland.
The speaker points out how you cannot go anywhere in Ireland without seeing lots of beggars. Many of the beggars are women with children.
Because of this, he concludes that there are too many people in Ireland. He argues (satirically, of course) that the best way to solve this problem is for the Irish to sell their children as food for rich people.