How do you save the game?
While Dreamscaper is a roguelite with permadeath, there is quite a lot of progress that persists. All Waking World progress gets saved automatically and Dream World progress such as collected crafting resources, defeated bosses and discovered memories get saved automatically as well.
You are also able to save your current run’s progress by exiting while in the Waking World. You can then pick up where you left off (including your entire inventory) by going to sleep.