In the heyday of sailing ships, all war ships and many freighters carried iron cannons. Those cannons fired round iron cannon balls. It was necessary to keep a good supply near the cannon, but how to prevent them from rolling about the deck? The best storage method devised was a square based pyramid with one ball on top, resting on four, resting on nine which rested on sixteen. Thus a supply of 30 cannon balls could be stacked in a small area right next to the cannon.
There was one problem… how to prevent the bottom layer from sliding or rolling from under the others.
The solution was a metal plate called a "Monkey" with 16 round indentations. But, if this plate was made of iron, the balls would quickly rust to it.
The solution to the rusting problem was to make "Brass Monkeys." Few landlubbers realize that brass contracts much more and much faster than iron when chilled.
Consequently, when the temperature dropped too far. the brass indentations would shrink so much that the iron cannon balls would come right off the monkey.
Thus, it was quite literally, "Cold enough to freeze the balls off a brass monkey".
And all this time, you thought that was an improper expression, didn’t you?