This is something you say to someone on their birthday. For example, in A. A. Milne's Winnie-the-Pooh,

``...[Is] it really your birthday?'' [Pooh] asked.

``It is.'' [Eeyore replied.]

``Oh! Well, Many happy returns of the day, Eeyore.''

Winnie-the-Pooh, Chapter 6.

"Many happy returns of the day" has nothing to do with returning presents, as I, at one time, thought. What it is, is a wish that "the day" (the birthday) returns many times in the future. In other words, it's a wish for a long and happy life.