Claudius probably suffered from birth defects. He apparently twitched, drooled and farted uncontrollably, which put a serious crimp in his social life.
But it is not likely these affected his intellect. Before becoming Emperor he was a well-respected historian, writing a history of the Etruscans and the only known multi-volume history of Carthage, which unfortunately have not survived. However, there are also no surviving claims that these were not useful works. He was shunted aside in the scramble for power largely because he started writing a history of the Civil Wars that was less than entirely laudable of Augustus.
Idiots don’t write respected works of history. At least not without ghost-writers, which seem to be a modern invention.
All of his relatives had died and/or been poisoned or murdered, notably his brother Germanicus, in the faction fights of the imperial family. It is likely he exaggerated his disabilities and functioned as something of a court jester to his nephew Emperor Gaius Caligula.
When Caligula and his wife and daughter were murdered, Claudius was the only survivor of the family and was acclaimed emperor primarily for lack of an alternative. He was probably the best emperor of his family after Augustus and possibly Tiberius, although this was admittedly a pretty low bar to jump.
The portrayal of Claudius in I, Claudius by Robert Graves and in its BBC dramatization is compelling. But it should be remembered that this characterization is a heavily fictionalized retelling, and is not necessarily historical truth.
Robert Graves' novels about Claudius draw heavily from Suetonius' Life of Claudius, I think.