When out in the Remnant Wasteland, some enemies will drop a unappraised item.
NOTE: I found the best way to farm these was by killing the Unique machine at the Floating Tower, and the Wanderer Eater which is located at the left as you leave the Extinct Civilization City, they might not always drop it but I had much better luck killing these for the drops. Unappraised items this can be anything from weapons to accessories. Once you have a few or a bunch go back to the main city and go to the entrance or behind home icons in the Urban District and go to Kirito's Room. When it loads your see Agil in the back left, approach him and select the "Can you appraise an item?" On the next menu select
He does "charge a small fee" this will relate to how the item will come out so some can be expensive or cheap.
The dearest item I had to pay to have appraised was 18,000 C and cheapest 4,900 C. So make sure you have some money before getting your items appraised. I can confirm I did multiple unappraised items at different times so you don't need 15 all in one go.
Upon having your 15th item appraised your achievement should unlock.