Apparently this was THE question New Yorker staff members and contributors were asked all over the world when the subject of the magazine came up in conversation. Biographer Linda H Davis gives us the long form answer in her new book Charles Addams: A Cartoonist's Life (link).
I actually loathe the Addams Family TV show and the Hollywood movies, both make a mockery of Addams' sensibilities - its like they set out to make weird cute. The real body of work is the cartoons and Linda Davis uses them throughout her wonderful biography to illustrate what it was like for Charles Addams to live true to his love of the strange, grim and grisly. My personal favorite cartoon (a copy hangs in my bathroom) is the one where two men encounter a topless mermaid on the beach, one whispers to the other: "Now, remember to act casual!"
Davis relates how Adams was once asked where he picked up "his interest in the bizarre" - his answer "that's like asking a bird how he learned to sing" If you come to this book hoping to find the secret key to all the weirdness you will be disappointed (although his childhood dentist did hang himself in the swamp!) But Linda Davis does not disappoint when it comes to telling the story of how Charles Addams made it big (I am not even going to go into the bit about the ladies here) and remained true to his artistic sensibilities.