Richard Connell's short story "The Most Dangerous Game" tells the tale of a man (Rainsford) who becomes trapped upon an island by a man (Zaroff) who has found, and hunts, the most dangerous prey in the world: man (given man is the only creature who can reason).
Rainsford is challenged by Zaroff to survive a three day hunt. If Rainsford is able to elude Zaroff, he wins the game. To this point, Zaroff has always won.
In regards to the quote in question, it comes from the following passage:
He knew his pursuer was coming; he heard the padding sound of feet on the soft earth, and the night breeze brought him the perfume of the general's cigarette. It seemed to Rainsford that the general was coming with unusual swiftness; he was not feeling his way along, foot by foot. Rainsford, crouching there, could not see the general, nor could he see the pit. He lived a year in a minute.
What this refers to is that when people are close to death, their life has been said to flash before their eyes. Therefore, in one sense, Rainsford may have believed that he was about to die.
On the other hand, the quote could be referencing the anxiety and anguish Rainsford is feeling as he is waiting for the trap to spring. For him, the time he spent waiting seemed like it lasted forever, meaning it felt like an entire year passed as he waited for the Burmese tiger pit to take its victim.