They are simply hot dogs made with molded Krabby Patty meat as the sausages. Mustard is added to them as a condiment. They are served in a red and white checkerboard paper food tray.
Role in episode
While making a batch of Krabby Patties, SpongeBob is fooling around with the extra meat. He molds it into the shape of a sausage, cooks it, and introduces it to Mr. Krabs. When SpongeBob shows Mr. Krabs how he cooked it, Incidental 42 comes in to look at where the employees are. He sees it as being new to the menu and spreads the word about it.
In no time, there are many orders, enough to prevent the customers from eating Krabby Patties. Mr. Krabs crosses Krabby Patties off the menu and takes away SpongeBob's spatula and grill, much to the latter's devastation. He even renames the restaurant the "Krusty Dogs."
Then SpongeBob and Squidward plan to get the customers to desert the restaurant to get Krabby Patties back. They are successful, but Squidward is still humiliated about the mascot costumes, as he later has to wear a Krabby Patty one and ends up getting even more humiliated.
SpongeBob becomes shocked when Mr. Krabs says that the Krabby Patties are gone. After many plans, he thinks of a way to get them back: He over-inflates one and the customers think it is an explosive. The customers then demand that Krabby Patties be brought back, which Mr. Krabs does, and the restaurant goes back to normal and Krusty Dogs are discontinued at the Krusty Krab.
This is not the first hot dog to be sold by the Krusty Krab. The first was the Krusty Kelp Dog, which appears in the episode "Nasty Patty."