Green’s Dictionary of Slang

apartment(s) to let phr.

also attic for rent, attic to let, space...
[the image is of a certain emptiness in the upper storey under upper adj.]

unhinged, insane, crazy.

[UK]Hotten Sl. Dict. 67: ‘APARTMENTS TO LET,’ said of one who has a somewhat empty head.
[UK]Sl. Dict.
[UK]C. Hindley Vocab. and Gloss. in True Hist. of Tom and Jerry 154: It is said that the ‘widows cap,’ denotes there are Apartments to Let! Also said of an empty-headed or shallow-cove.
[US]Van Loan ‘Levelling with Elisha’ in Old Man Curry 35: They ought to have a look at his head. He’s got an attic for rent, sure.
[Aus]Sun. Times (Perth) 9 Mar. 7/4: Another very effective way he has of quelling the troublesome one is to tap the head while saying very solemnly [...] ‘There’s nobody home,’ or ‘Apartments to let’.
[US]Maines & Grant Wise-crack Dict. 14/2: Space to let – A vacant stare.
[UK](ref. to 1870) Dundee Courier 6 Apr. 6/3: Have we lost the art of slang? [...] It did not pay to be stupid in those days [...] you had ‘apartments to let’ or [you were] chuckle-headed, buffle-headed, cabbage-headed [and] chowder-headed.
[US]H. Rawson Dict. of Invective (1991) 344: with an apartment to let; with an unfurnished attic (or garret); with a room for rent.