If you want to see some past cutscenes involving Aymeric, the "Unending Journey" you see in all inn rooms allows you to view some (but not all, annoyingly enough).

If you don't mind reading, try these: https://ffxiv.gamerescape.com/wiki/C...ger_Transcript

Cross-reference against your own in-game journal to determine which quests would've included a scene with Aymeric speaking, then search away for that quest in the Loremonger.

Lastly, don't forget to thank Moose.