Throughout the 1800s the United States of America had copyright law, but refused to recognize copyright established in other countries. There was, in effect, legally-permitted pirating of all foreign written works, including the works of popular authors like Maria Edgeworth, Benjamin Disraeli, and Charles Dickens. This continued until 1896 (and in fact, a bit after), when the US Congress joined the international copyright union, officially making this piracy illegal.
Established publishing houses took it upon themselves to fight this piracy, although they did it not from noble ideals, but rather because it gave them a good chance to turn a profit. By making a deal with the author to receive a copy of the manuscript at the same time as the publishers in England, they could release at the same time and thus be on the shelves before a pirating press had anything to copy. This gave them a chance to collect a good portion of the market share, and kept publishing the works of foreign authors profitable.
Of course, once the copies were on sale, a pirating press could rush to set type and go to print themselves. This was not a quick and easy process in those days; professor Stephen Breyer estimated that a pirate copy could take 6-8 weeks to set and publish.
These pirated copies might not have been a big threat to the American publisher, but now this was a matter of principle! Now someone was stealing from an American publisher! Well, this wasn't enough to get the government involved, but it required a response. Hence, fighting editions.
Once a pirated copy came out, the original (American) publisher would come out with a even cheaper edition, the fighting edition, to undersell the pirated copies and keep the competing press from making a profit. This practice was common enough that it was not always necessary; the threat of a publisher bringing out a fighting edition was enough to stop some pirating presses from publishing at least some works.
It appears that fighting editions primarily appeared in the arena of the 'respectable' publishing industry, and that these larger presses pretty much ignored the dime novels that might at times copy a well-known work. This is probably due in part to market segmentation -- there's no real profit to be had selling 4ยข pamphlets to the working class -- and partly due to the dime novels mostly preferring lowbrow blood-and-thunder over more reputable (and costly) works.