There's a scene in Wes Anderson's The Royal Tenenbaums where Gene Hackman is a dad visiting the family from which he has become estranged to see his precocious children perform a play in which they all dress up as animals. Asked afterwards for a verdict by his daughter Margot, who produced it and cast herself as a zebra, he complains: "It didn't seem believable to me. It was just a bunch of little kids dressed in animal costumes." Needless to say, he was not invited back.

With Fantastic Mr Fox – Anderson's wickedly enchanting animated adaptation of a Roald Dahl children's story about a big-headed fox cheekily going to war with local chicken farmers who object to him culling their henhouses – little Margot is vindicated. Puppet animals running around in jerky stop-motion animation might seem even less real than Margot's childish dressing-up charade, yet Anderson brings their hand-made world hilariously alive through sheer visual sleight of hand.

Of course, it helps having the fox smartly voiced by George Clooney, with Meryl Streep as his long-suffering wife and Jason Schwartzman their self-absorbed cub who's always butting in. Bill Murray, a regular in Anderson's films since Rushmore, is an avuncular badger, and when a rat strikes up on a banjo, it's Jarvis Cocker.

Anderson was born in Houston, Texas in 1969, the same year Dahl wrote Fantastic Mr Fox. "We grew up on his stories," he says. "We read everything he wrote. We wanted to be him." The film has just opened London Film Festival – repeating the achievement of Up in opening Cannes Festival – so we meet in the old-world Lanesboro Hotel. A familiar figure in lightweight pale brown suit and white open-necked shirt, he sits under a chandelier in an antique chair with intricately carved wooden arms in the form of snakes that could be a prop in one of his films. The son of an advertising executive father and archaeologist mother who divorced when he was seven – a family not dissimilar to the Tenenbaums, "although the father is nothing like my father" – he, like Margot, wrote plays for his siblings and later for schoolfriends at St John's in Houston, much in the manner of Max, the central character in his breakthrough film Rushmore. "I can't think of anything that wasn't plagiarised. Everything I tried to write in those days could be traced to a source. I've always wanted to do stories like the stories and films I loved as a child."

Anderson was already discussing a film version of Fantastic Mr Fox with Dahl's widow Liccy in 2000 before he even began The Royal Tenenbaums, and he went on to make The Darjeeling Limited and The Life Aquatic With Steve Zissou before he managed to get back to it. He employed animation elements in all three films. The Royal Tenenbaums is broken into chapters, each introduced in the form of captioned illustrations, while The Life Aquatic has a stop-motion monster shark. Going the whole hog and making a fully-animated feature was the logical next step.

"What I love about stop-motion is that you can sense there are people's hands making these things and moving them around. It just feels personal, and there's some charm about that, some sort of extra life that goes into it."

The fact that it's time-consuming didn't deter him. He spent twelve months shooting Fantastic Mr Fox at 3 Mills Studios in London, carefully assembling the 62,000 frames required to animate the story. "There's the same sort of pressure to get it finished within budget and time constraints as there is on other films, but it's a much longer time period so you have a sort of slow-motion anxiety continually going on. You've no choice other than to design each object. Absolutely everything filmed has to be manufactured because it's miniature. You can't just bring things in." Although the story is fantasy – a black comedy farmers-versus-vermin variation on Watership Down – the quaint hand-made authenticity of the décor makes it human and believable. Even the tiny teacups are exquisitely crafted. "I tend to be a bit old-fashioned. I'm most drawn to old things. I was sad when films shifted to working on computers for editing. It was so nice to see the strip of film and actually touch the celluloid."

The Fox household with its meticulously detailed furnishing was inspired by Dahl's actual home in Buckinghamshire, where Anderson worked with co-writer Noah Baumbach on the screenplay. "It led us to basing the character more on Dahl than it was in the book. The design of the film became his house and the world around it."

The sound was recorded on location rather than in padded studio booths. "I wanted to hear the wind blowing in the trees, the voices echoing in the room." Clooney and most of the voice cast holed up in a farmhouse where they ate all their meals together, just as everyone shared the house of the characters in The Royal Tenenbaums, and chuffed across India together in a train in The Darjeeling Limited. "I wanted to make it more of an adventure so it's fun to do. The thing I didn't expect was that they would really act out their scenes, running around fields and jumping over things."

Meryl Streep recorded her scenes in a studio in Paris, but not a voice studio. "It was for orchestras so it was a great big room. We had a little area in the middle with a couch and some armchairs and a rug on the floor, so she was free to act out her lines too."

There's a popular misunderstanding that animation is fantasy while live action is real: in fact, all film is artifice. Anderson's originality is in finding a quirky space in between. Nor does he regard animation as something just for children. Dahl wrote children's stories and adult stories and often combined the two.

"There's nothing that makes them forbidden for children and nothing that excludes adults. I'm inspired in the same way by the cartoons of Hayao Miyazaki, particularly Spirited Away. Of course a lot of animation in Japan is not for children at all. I'm not sure what my next film will be, but someday I'd like to do an animated film that's not intended for families, just for adults."

Fantastic Mr Fox opens on Friday