The Memory Fragments you collect are listed in the "Memory" menu.
And some, for whatever reason, come through garbled and unreadable.

Each memory carries stars for how hard it is to obtain.
The more stars, the bigger the bonus for viewing it. But...

Some bonuses, though rarely, raise a character's Lv, or shift the Ideal Value (IV) that gives form to their true essence, and the way you fought until then changes with it. A memory you have come to know, and the power it gave you, are engraved in that person forever, as part of who they are. You cannot forget.