What does "Many, many happy returns of the day" mean in context of a birthday?

Part of the phrase has been truncated. The full statement would read:

"[I hope you have] many happy returns of the day."

"The day" is your birthday, and if it returns "many times" it means that you will live to be quite an old person.

Wikipedia explains the phrase as follows:

The term itself refers to the passing year. Since the 18th century this has been used as a salutation to offer the hope that a happy day being marked would recur many more times. It is now primarily used on birthdays; prior to the mid 19th century it was used more generally, at any celebratory or festive event. Current usage is often as a more formal option than 'Happy Birthday'. It is also often to be found on greetings cards.

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An alternative explanation is that "returns" here is used in the sense of "yield" or "profit" that it is still found in "investment returns". Therefore "many happy returns of the day" would be a wishing a person a rewarding day, full of happiness.

The first explanation implies we are wishing that their birthday will return many more times - in other words, "May you live long".