Chance (Peter Sellers) is a middle-aged man who lives in the townhouse of an old, wealthy man in Washington, D.C. He is simple-minded and has lived there his whole life, tending the garden. Other than gardening, his knowledge is derived entirely from what he sees on television. When his benefactor dies, Chance naively says to the estate attorney that he has no claim against the estate, and is ordered to move out. Thus he discovers the outside world for the first time.
Chance wanders aimlessly. He passes by a TV shop and sees himself captured by a camera in the shop window. Entranced, he steps backward off the sidewalk and is struck by a chauffeured car owned by Ben Rand (Melvyn Douglas), an elderly business mogul. In the back seat of the car sits Rand's much younger wife, Eve (Shirley MacLaine).
Eve brings Chance to their home to recover. Drinking alcohol for the first time during the car ride home, Chance coughs as he tells Eve his name. Eve mishears "Chance, the gardener" as "Chauncey Gardiner". Chauncey is wearing expensive tailored clothes from the 1920s and '30s, which his benefactor had allowed him to take from the attic, and his manners are old-fashioned and courtly. When Ben Rand meets him, he assumes from these signs that Chauncey is an upper-class, highly educated businessman. Chauncey's simple words, spoken often due to confusion or to a stating of the obvious, are repeatedly misunderstood as profound; in particular, his simplistic utterances about gardens and the weather are interpreted as allegorical statements about business and the state of the economy. Rand admires him, finding him direct and insightful.
Rand is also a confidant and adviser to the President of the United States (Jack Warden), whom he introduces to "Chauncey". The President likewise interprets Chauncey's remarks about the "garden" as economic and political advice. Chance, as Chauncey Gardiner, quickly rises to national public prominence. After his appearance on a television talk show, he becomes a celebrity and soon rises to the top of Washington society. He remains very mysterious, as the Secret Service is unable to find any background information about him. Public opinion polls start to reflect just how much his "simple brand of wisdom" resonates with the jaded American public.
Rand, who is dying of aplastic anemia, encourages Eve to become close to Chauncey. She is already attracted to him and makes a sexual advance. Chauncey has no interest in or knowledge of sex, but mimics a kissing scene from the 1968 film The Thomas Crown Affair, which happens to be on the TV at the moment. When the TV scene ends, Chauncey stops suddenly and Eve is confused. She asks what he likes, meaning sexually; he replies "I like to watch", meaning television. She is momentarily taken aback, but decides she is willing to masturbate for his voyeuristic pleasure. As she becomes involved in the act, she does not notice that he has turned back to the TV and is watching it, not her.
Chauncey is present at Rand's death, after which he talks briefly with Rand's physician, Dr. Allenby. During their conversation, Allenby realizes the truth â that Chauncey is merely a simpleminded gardener who knows nothing of finance or politics â but does not appear bothered by it. At Rand's funeral, while the President delivers a speech, the pall-bearers hold a whispered discussion over potential replacements for the President in the next term of office. As Rand's coffin is about to be interred in the family mausoleum, they unanimously agree on "Chauncey Gardiner".
Oblivious to all this, Chauncey wanders through Rand's wintry estate. He straightens out a pine sapling and then walks off across the surface of a small lake. He pauses, dips his umbrella into the deep water under his feet as if testing its depth, turns, and then continues to walk on the water as the President quotes Rand: "Life is a state of mind."

Question: What did Allenby realize after speaking to Chauncey?
Answer: He realized that Chance was a simple gardener.
Alternative answer: He was actually not very intelligent