In this chapter, Pap is drunk and disgruntled (a common condition for him) because he spent last night drinking and lying in the gutter. He is a mess in every way and, instead of blaming himself for his pitiful condition, he needs to find something or someone else to blame for his circumstances. In this case, he blames the government. The complete quote is as follows:
"Oh, yes, this is a wonderful govment, wonderful. Why, looky here. There was a free nigger there from Ohio--a mulatter, most as white as a white man. He had the whitest shirt on you ever see, too, and the shiniest hat; and there ain't a man in that town that's got as fine clothes as what he had; and he had a gold watch and chain, and a silver-headed cane--the awfulest old gray-headed nabob in the State. And what do you think? They said he was a p'fessor in a college, and could talk all kinds of languages, and knowed everything. And that ain't the wust. They said he could VOTE when he was at home. Well, that let me out. Thinks I, what is the country a-coming to?"
His argument is that it is the government which allows a free slave to have a much better life than he has. This free slave is a very intelligent college professor who dresses in fine clothes and can speak several languages--and the final insult is that he can vote. This is just more than Pap can take, and of course he has to blame someone. He blames the government for allowing those things while the illiterate Pap is dressed in muddy rags and sleeping off a drunk in the gutter. He disdains this unnamed black man for being everything he is not and ironically blames the government for creating an unjust world in which a freed slave has a better life than he does.
Twain's satirical intent is to link the government to something so obviously not in the government's control--what a person decides to do with his life. The only thing the two men have in common is their ability to vote, and Pap refuses to vote at all if such a man as this gets to vote, too. It is an unreasonable position from an unreasonable man, and Twain's satirical purpose (making fun of something in hopes of changing it) is to remind his readers that it is not the government but our own choices which determine our positions in life.