Bootable games had to implement their own disk-handling routines (on top of the BIOS), if only to read the data from their disk(s) in order to boot and run in the first place. If you can read from a disk, it’s not much harder to deal with writes, especially if you use a fixed-layout disk format. The BIOS itself provides services to read from and write to disks, and it’s reasonably straightforward to use those to fulfill a bootable game’s requirements. Bootable games which needed to store data would write it to their own floppy disk, or sometimes to another floppy disk, not to a hard drive — many users of bootable games wouldn’t have had a hard drive anyway. (This data wasn’t necessarily players’ progress; often it would be high scores, as in for example Windmill Software’s bootable games.)
As far as the why is concerned, there are two main reasons I’m aware of: the first is that by skipping the operating system, the games had more memory available to them; and the second is that by using game-specific disk formats, they were slightly harder to copy, modify etc. (DOS couldn’t read the floppies.) Many if not most bootable PC games were copy-protected in one way or another.
The Digger remastered web site has loads of information on that particular bootable game and a few others. Flopper is a nice little tool which can boot many such games using disk images rather than real floppies. The best overview of PC bootable games is Retrograde Station (which now only exists as preserved on the Internet Archive, or partially mirrored on the Wizardry Fan Page). Nerdly Pleasures also covers a number of PC booters (including one non-game, Will Harvey’s Music Construction Set) and PC floppy disk copy protection.