Old guy literary critic and celebrated scholar Harold Bloom coined this phrase in his 1973 book of the same name. According to Bloom, the "anxiety of influence" describes the way poets feel about those who have written before, particularly in their culture or tradition. That is, poets writing today are inspired by the literature of the past, sure, but they're also afraid of just repeating what's already been said. Today's poets are worried about not being original.
Word (okay, words) of warning: though Bloom used the word "poets," he's usually specifically referring to the big white male poets from the Western canon. Just sayin'.