When Sanger Rainsford enters the home of General Zaroff, he asks his host if Ivan is Russian. Zaroff tells Rainsford that he and Ivan are both Cossacks.
Historically, the Cossacks were an ethnic group centered in southern Russia and southeastern Ukraine and noted for their military prowess and excellent horsemanship. The etymology of the word "Cossack" is free man or adventurer, which is fitting for Zaroff, the master of the hunt. It is clear that Zaroff is proud of his heritage.
In Russia and nearby areas, borscht is traditionally served at wakes after a funeral has been held. Because it is made with beets, it is bright-red in color. Zaroff and Rainsford eat borscht together, and this could be symbolic in a few ways. First of all, it is a way for Zaroff to celebrate his cultural identity as an Eastern European. It is clearly a luxury, since the ingredients would have to be imported, and so it is yet one more way for Zaroff to flaunt his wealth.
Because the dish has an association with funerals, it could be a bit of foreshadowing, since only one of the two men will emerge alive from the hunt that Zaroff has planned for the following day. Or, it could be a private joke between the two Cossacks, since they both fully expect that Rainsford will be dead soon.
And finally, the soup's rich red color could suggest a connotation of blood, as these two world-class hunters dine together and discuss their hunting exploits. It also foreshadows the blood that will be spilled the next day.