You save through the save menu, bottom of the screen.
Sometimes, it stops working.
It always did for me in the first scene with the ... young demon or something?
Anyways, the "story" is mindboggling. The whole thing looks like an inhomogeneous amalgamation of various anime tropes with a horrible translation, directed by a 15 year old. Or 13. I hope it doesn't take itself seriously.
It could come together nicely, given time, but now is not yet that time. On the plus side, the graphics are ok, if one discounts the 2000s console UI with overly large buttons and glow effects.
Of note, I had to sleep twice to progress. The first time just got me from day 0 to day 1, but didn't fulfil the quest to sleep.