The OED defines this useful phrase as meaning “a situation or statement has at last been understood; a person has reacted belatedly” and notes, “Originally used with allusion to the mechanism of a penny-in-the-slot machine.” The first citation is from The Daily Mirror in 1939; all the later cites are British as well.
As I say, it’s useful phrase. The closest American equivalent would be something like, “The lightbulb went on,” which, besides being clunky, lacks the apt imagined “click” of the penny equivalent.
In addition to the OED cites, this Google Ngram Viewer chart suggests the phrase is indeed of British origin. The red line indicates British usage, the blue American.
(There’s a fair amount of noise in the chart, emanating from references to actual pennies actually dropping.)
The chart indicates a steady rise in U.S. uses through 2008, and it appears to be continuing. In the New York Times, through 2006, the phrase almost always appeared as part of quote by a British or Canadian person. But there have been about fifteen uses of it by Times writers since then. Quite a few of them came from the pen of one person, Deb Amlen, who writes the “Wordplay” crossword puzzle blog. Clearly, pennies have to drop or the puzzle doesn’t get solved.