William Faulkner gave his Nobel prize acceptance speech in December 10, 1950. This period was only a few years after World War II had ended. On Faulkner's mind were the devastating bombing of Germany with conventional weapons and the use by the United States of atomic bombs on Hiroshima and Nagasaki. The Soviet Union was developing nuclear weapons as well, and facing off against the United States in what was called the Cold War. One central doctrine of the Cold War was known, with one of the most appropriate acronyms of all times, as MAD, "Mutually Assured Destruction". This was the idea that if both superpowers kept around enough weapons to exterminate all life on their enemies' continents, no one would actually dare to use these weapons. Thus, when Faulkner talks about living in fear, he is talking about an era in which many people were genuinely afraid that the world would be destroyed in a nuclear Armegeddon.
Faulkner is arguing that writers need to focus on the truths of the human heart to get their job done as novelists and poets, and that the fear of nuclear destruction makes this difficult or impossible.