I was wondering what "drop the penny", "help get the penny to drop", or things similar mean? All I can understand is that it must be a metaphor. For example:
simply trying to repeat things in perhaps a slightly different way may help get the penny to drop (think about the student coming to office hours who has read the book but doesn't get it; just going over it again in different words can help).
How to use such an expression in life generally. Can you give other examples?
I think this comes from old-fashioned pay phones (or any other vending-type machine, like a drugstore gumball machine), where sometimes you'd put the money in and it'd get stuck, so you'd have to try various strategies (generally consisting of variations on "hit the dang thing") until the coin would finally get dislodged and drop, allowing you to make your call (or get your candy). The metaphorical meaning is thus to understand a particular bit of information after repeated failure. To use another metaphor, it's to finally get that lightbulb to go off.
Martha has the right idea on the metaphor 'the penny drops' is from the situation in coin-operated machines where the "penny drops" engaging the function of the machine. As coins often get stuck, then suddenly become unstuck, this term came in to general use for the situation where people were mentally stuck with a concept and then it became clear.
See the following link: http://www.phrases.org.uk/meanings/280900.html
I had a tour of the caves underneath Nottingham Town centre where people lived, tanners worked and people sheltered during the war.. Above one of the caves was a pub. between the pub and cave was a little hole leading between. If the police turned up, to warn the unsavory goings on downstairs in the cave the landlord would drop the penny thro the hole hence the name the penny has dropped
