What inspired C.S. Lewis to write the Narnia series?

There is no one answer to this question - each book had a different reason for being written, and he wrote more than 30 books plus many letters, essays and poems. If you are referring to The Chronicles of Narnia, he said that the book series originally started with a series of pictures that he had in his head. He describes this process in an essay entitled "Sometimes Fairy Stories May Say What's Best To Be Said", published in "Of This and Other Worlds".