Mr. Pickwick was a central character in The Posthumous Papers of the Pickwick Club by Charles Dickens. Mr. Pickwick is a rather dotty, but good-hearted old man. And so:
Pickwickian most obviously means someone who is naive, kind and generous, and yes, dotty. A Pickwickian old man.
But it is more often used to mean 'used in an unusual manner,' or 'not to be taken literally, especially when referring to words and phrases. He's using the word in a Pickwickian sense. A quick Google search also finds references to "Pickwickian parts", "Pickwickian precedence", and "Pickwickian skills", suggesting that common usage has generalized the word to mean anything unusual or odd.
This second sense of Pickwickian comes from the book itself. The club members call them selves Pickwickians, and in the first chapter of the book (written in the style of minutes from a club meeting) Mr. Blotton tries to take back an insult by claiming that he had intended it in a special, 'Pickwickian" sense.
The Chairman felt it his imperative duty to demand of the
honourable gentleman, whether he had used the expression which
had just escaped him in a common sense.
Mr. Blotton had no hesitation in saying that he had not -- he
had used the word in its Pickwickian sense. (Hear, hear.) He was
bound to acknowledge that, personally, he entertained the
highest regard and esteem for the honourable gentleman; he had
merely considered him a humbug in a Pickwickian point of view.
(Hear, hear.)
Mr. Pickwick felt much gratified by the fair, candid, and full
explanation of his honourable friend. He begged it to be at once
understood, that his own observations had been merely intended
to bear a Pickwickian construction. (Cheers.)
There is also a medical condition, Pickwickian syndrome, named after Mr. Pickwick, although it wasn't Mr. Pickwick but another character in the book who was afflicted.
The word wellerism also comes from The Pickwick Papers.