I got this right out of the guide

"The Bestiary is an equippable item that can be obtained by trading the Small Ring found in the Desert Temple to the redheaded girl up by the northwest house of Gatero. It works much like a sword, in that you use it to physically strike enemies.
Striking the same enemy a set number of times causes that enemy to be recorded within the Bestiary. The number of hits required varies by monster. When you've satisfied the hit requirement, a sound will play and the word "GET" will appear on screen to let you know you've got it. Once you've recorded a certain monster, you can view its info by selecting the Bestiary option from the Start menu. Use L1 and R1 to scroll through pages, or through monster entries if you're viewing one."