Usually those who play it make a save right at the end of the update (before the "thank you for playing" screen). Saves are stored in two folders: the save folder in Eternum's game folder, and the other is somewhere in the Appdata folder. Because the latter exists, it's safe for people to simply delete the game when they're done if they need the space, since the saves will still be accessible once a new update comes along.
When an update gets uploaded (either a leak for the Demon Lord tier or any of the other patreon tiers or the Public Release), players are basically redownloading the game with the latest update already patched in. As of now there are no "Patch Only" uploads for the game where the player can download the patch and then apply it to their copy of a previous version of the game (which saves bandwidth). And no, I don't know why that is, maybe Cari just finds the process tedious on his end, which is why you and your bandwidth will just have to bear with it.
Once you DL the game with the latest version, you can load from the save prior to the "thank you for playing" screen and the game just continues from there (unless you've played the game with a modded save, in which case you're gonna have problems unless you apply the latest version of the mod you were using, one that's compatible with the latest update).
Sidenote though: if you're playing a leaked Demon Lord beta version, then expect occassional crashes when you try to load saves and typos in the game's narrative and dialogue. If you want a more polished experience, it's recommended that you wait for the Public Release of the latest update as that's usually the version where the bugs are ironed out and the typos have been proofread.
Hope that covers everything.