wanted to save a backup of the game but the download doesn't start

I'm not sure I understand your question, so please correct me if I guess wrong.

I think you are trying to download the offline installers. And when you go to your game library and select Encased. Then under "DOWNLOAD OFFLINE BACKUP GAME INSTALLERS" you click "Encased: A Sci-Fi Post-Apocalyptic RPG (Part 1 of 5)" and nothing happens? If this is correct, then I can tell you it works fine for me. When I tried it just now, it downloaded the small (1 MB) file almost instantly. (of course, I would need also to download the other 4 files for the task to be complete)

Can I ask you which browser you are using? In my browser, Chrome, I can click the 3 dots in the upper right-hand corner, then select "Downloads" from the drop-down menu. Then you should see a list of the files that have been downloaded. You should find the file you downloaded in this list.

(for me, this last part is a little different because I have a 3rd party download manager that takes over this task, so in my case I have two places to look for the file: 1) in my browser list of downloaded files as described above and 2) I have to launch 'Internet Download Manager' and look there as well.)