It means that you are showing how old you are--or showing that you're old(er). It is usually used by older people, or at least by people who are older than the person with whom they are talking.
For example, if 31-year-old me made a reference to a '90s TV show while talking with my 20-year-old student, I might follow it up by saying "I'm aging myself".
I hope that makes sense! I would be happy to explain further, in English or in French!