It means that some time has passed since the original comment and it has now become clear that the comment was incorrect, badly conceived or inappropriate in some way. It is meant in a humorous way.

Last year, your best friend had a big promotion. You complimented him, "Hey bud, you're the man." This year he had a gender reassignment surgery. Well, your compliment didn't age too well.

You say this phrase when something happens after the comment was made that make the comment seem wrong, stupid or silly in retrospect. For example, if I write on Twitter that "Joe will never become a millionaire" and then, a week later, Joe later wins the lottery and becomes a millionaire, I can say, in reference to the statement "Joe will never become a millionaire", that the comment did not age too well.