In Swift's "A Modest Proposal," when the speaker says that
I shall now therefore humbly propose my own thoughts, which I hope will not be liable to the least objection....
he, first of all, is literally saying that he hopes no one will object to his proposal. This makes sense because he is proposing something that he supposedly wants others to approve of. He would naturally want everyone to agree.
Figuratively, of course, the speaker is being ironic. He says he hopes no one will object, because he knows everyone will object. He uses this kind of irony throughout the essay, starting with the title: his proposal is anything but modest.
That's the long, official answer. The short answer is that he's being funny; making a joke.
In my opinion, the author puts this phrase in as a way of making his proposal even more satirical.
In this essay, one of the things that makes it so effective is how deadpan the author is through the whole thing. There is not really any place in the essay where he makes it look like he's joking. So he's very seriously putting forward this proposal that is just totally outrageous.
I think that the line you mention helps him do this. It's like he's starting out saying "this is totally serious, no one could possibly object to this."
So, I think he says it to make it sound like he's more serious so that his satire will be more effective.