Just to refine Bob's answer slightly, it is ONLY used at birthdays. We would never use it in any other situation so it's not really a substitute for "congratulations" if someone has passed an exam, got engaged, or won the Republican nomination for president. It can also be shortened to "many happy returns"

It's a British saying that means more or less "Congratulations". It is often used instead of Happy Birthday.

It is used in the United States. It's usually used for birthdays. The key is that it is a situation that deserves congratulations, but is expected to repeat annually. For example, if your birthday falls on April 1st, then the day--April 1st--"returns" over and over again: April 1st, 2016; April 1st, 2017; April 1st, 2018. A wish for "many happy returns of the day" is thus a wish for a long life (many returns) and a happy life (many happy returns).