The incident concerning the Grangerfords occurs in the novel in Chapters 17 and 18. The two central aspects that Twain is appearing to be satirising in this section is the mawkish preoccupation with death as expressed through the character of Emmeline Grangerford and then the hypocrisy of the Grangerfords and the Shepherdsons in their blood fued.
Firstly, Emmeline Grangerford, although she is now dead, is presented as a young lady with an unhealthy interest in death. It is hilarious the way that we are told she was so interested in death that she was famed for arriving to give her respects even before the undertaker:
Every time a man died, or a woman died, or a child died, she would be on hand with her "tribute" before he was cold. She called them tributes. The neighbours said it was the doctor first, then Emmeline, then the undertaker - the undertaker never got in ahead of Emmeline but once, and then she hung fire on a rhyme for the dead person's name, which was Whistler.
Her inability to find a rhyme in time for "Whistler" is what drives her to her grave, as she let the undertaker "beat" her to the dead person's family. This is clearly ludicrous and shows Twain at his best poking fun at those with an unhealthy interest in death.
Another aspect that is satirised is the feud between the Grangerfords and the Shepherdons, which has gone on so long that no one remembers what started it. In Chapter 18, Colonel Grangerford is presented as a civilised man, and yet there is an amusing incident when they go to church and everyone takes their guns and keeps them between their knees during the service whilst they listen to a sermon on "brotherly love." Although they talk about it on the way back, it clearly has no impact on them in terms of the rancour between the two different families.
Twain therefore seems to be using the Grangerfords to satirise the hypocrisy of the Southern Aristocracy and the unhealthy interest in death expressed by some women. It would be interesting to find out what he would think of the poetry of Emily Dickinson, who some critics have argued is similar to Emmeline Grangerford!