General Zaroff is able to finance his lifestyle because he invested heavily in American securities. Many of his countrymen had lost everything they had due to the 1917 Russian Revolution and the overthrowing of the Czar, but he was lucky that he thought ahead and left the country when he did, so he could support such a lavish way of life.
In the exchange between Zaroff and Rainsford, we learn that Zaroff's father was a wealthy man: "He was a very rich man with a quarter of a million acres in the Crimea... ." So, readers can infer that part of Zaroff's financial success is partly because he had a wealthy father who probably passed on his wealth to his son.
General Zaroff was also an "officer of the Czar"--meaning he was a military man. This is also a source of income.
Zaroff also says that after the Russian Revolution of 1917 when communists overthrew the czar, he had to leave: "Many noble Russians lost everything. I, luckily, had invested heavily in American securities." Zaroff associates himself with noble Russians, making readers assume that he is of a higher, wealthier class. But instead of losing his fortunes, he invested in America's stock market which obviously paved a way for him to ensure financial stability.
Finally, when Rainsford is putting on clothes offered to him by Zaroff, he notices that the suit "came from a London tailor who ordinarily cut and sewed for none below the rank of duke." This tells the reader that Zaroff is simply no ordinary officer, but an official ranking officer with power. This substantiates some of his income as well.
Thus, we can infer that Zaroff has made money from his inheritance, the military, and investments.
Zaroff was a Russian nobleman but after the revolution was forced to flee. He then invested in American securities and became wealthy that way.