I was unable to find an interview or authorial statement that clearly spelled out Richard Connell's intent in writing "The Most Dangerous Game," so I will answer based on my interpretation.
It seems that the intent behind "The Most Dangerous Game" can be broadly separated into two catagories: to create a story of suspense that thrilled his audience; to explore the morality of hunting animals by comparing it to hunting humans.
The first point is clear in the story itself; it is, in fact, a thrilling story of suspense. For almost one hundred years, the story has influenced writers in film and print, and is still taught and read as a classic today.
The second is subtext in the story; while the author never digresses into a didactic monologue about the morality of hunting, the comparison of Rainsford and General Zaroff shows a clear question as to the morality of hunting as a sport rather than as a necessity of survival. Zaroff is bored with hunting animals (an accepted practice) and so he hunts humans (a condemned practice). Rainsford is of normal society, but he quickly puts aside his ethics and hunts Zaroff as a method of survival; Zaroff considers his actions acceptable although they would result in his own death, if they were successful. By the end, when Rainsford makes a conscious decision to kill Zaroff rather than attempt a hidden escape, it is unclear whether he has embraced Zaroff's philosophy, or killed Zaroff to stifle it forever.