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When dogs are used in hunting animals, they bay when they have cornered or treed their quarry. In this story, it represents the feeling that Rainsford has about not being able to escape his situation - he is forced to face the danger in front of him.
Cornered animals are also the most ferocious, so Rainsford is also saying it in the context of being willing to take on General Zaroff, having been left with no other alternative.
The line you mention is spoken by Sanger Rainsford to General Zaroff right at the end of the story. He is in Zaroff's bedroom -- he has managed to get there after escaping from Zaroff's efforts to hunt him.
In terms of hunting, an animal was said to be "at bay" when it had been chased (by dogs) until it had no where left to run. The animal would then have the choice to either fight or die (or both).
Rainsford says that he is "at bay" because those are his choices -- he is cornered and now he can fight or die or both. As it ends up, he fights Zaroff and kills him.
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