It was necessary to keep a good supply of cannon balls near the cannon
on old war ships. But how to prevent them from rolling about the deck was
the problem. The best storage method devised was to stack them as 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 only one problem -- how to prevent
the bottom layer from sliding/rolling from under the others. The solution was
a metal plate with 16 round indentations, called, for reasons unknown, a Monkey.
But if this plate were made of iron, the iron balls would quickly rust to it.
The solution to the rusting problem was to make them of brass - hence, 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 just a
vulgar expression, didn't you?