Credits to this site for the solution: tech-recipes.com
To save a replay of your last game, enter the Config menu, hold Up for a few seconds to enter the Kakusi menu, highlight "SAVE DEMO.REP" and press Z.
Replays are saved in the ETC folder in the installation folder as DEMO.REP. You will need to rename this file and/or copy it somewhere else in order to view/save additional replays.
To view a replay, place it in the ETC folder in the installation folder. Rename it to DEMO.REP if it is not already named as such. Then, launch the game and wait at the title screen until the attract mode starts; the replay should automatically load and start playing.