Yeah the link is broken, I had to google it.

The age estimates are based on having found a correlation between parental age and different types of mutation. The different "letters" of DNA are chemically different and the various kinds of single-letter swap mutations happen at different rates. They found by looking at present-day babies that babies with old mothers tend, on average, to have different kinds of new mutations to babies with old fathers. They then extrapolated this idea to the database of historical mutations.

How many thousands of years ago each mutation happened was estimated by a different team in a previous paper, also using statistical methods. I don't know the details of that part but I'm guessing if a gene variant is super common and widespread, it's probably old.

This all produces estimated averages for each era - they know they can't actually put an exact age and date on any one mutation, but they don't have to to get a rough average over thousands of mutations for each millennium.