The tattered moon page is a torn page obtained from monsters in the Forthos Dungeon. Four of these are used to fill the tome of the moon during the miniquest In Search of Knowledge, although players do not need to begin the miniquest to begin obtaining them.
Once the tome has been completed and given to Logosia in the Arceuus Library, subsequent pages can be given to her in exchange for 1,000 coins per page. Should players wish to stop receiving tattered pages from monsters in the Forthos Dungeon, they can speak to Logosia once they've completed In Search of Knowledge.
For an exhaustive list of all known sources for this item, see here