I think they may be referring to aging in the bottle/keg before you drink it.
Being new myself, I certainly am not meaning to be condescending, simply passing on what itty bit of knowledge I have gained.
The fermentation is different than the aging process. Fermentation for most 'regular' beers is usually only a couple weeks. There are guys that let it sit in the primary (first stage) fermenter for a lot longer, but that's a subject for many other threads. The aging process is when your 'wort' is done fermenting and then you put in the carbonating sugar (in whatever form that may be...me at my early pre-pubescent beer life, I am still using just plain 'ol corn sugar) and then transferring that into the bottle, capping it, and putting it away to do its thing at a certain temperature for a certain period of time.
A lot of the standard ales/lagers/etc. age about 30 days, maybe a bit longer....but I do know the stouts I have been researching talk about being better after aging for 3, 4 or even 6 months or longer.
tom_hampton and Maine Homebrewer have hammered me on being patient....not something we like to be when looking to enjoy the fruits of our labors, and it has paid off in spades. My beer has become MUCH better as I wait..... not that I don't cheat and snark one.....just to see how it's going....