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).