I don't know much in terms of details, but I'm pretty sure the basic process involves building parallel volume data structures and then switching which one's used.
To understand what that means, you need to know a little bit about how a filesystem stores its files on disk. Somewhat oversimplified, there are two kinds of data being stored: the actual file contents are just stored pretty much as raw data, and then there's some additional database-ish thing that keeps track, for each file, of where that particular file's raw data is stored (as well as "metadata" like file's name, permissions, last modify time, etc).
The raw file contents is pretty much the same between different volumes formats (except for encrypted volumes), but the database-ish metadata will be different between HFS+ vs. APFS vs. FAT vs. whatever.
Most of the disk's contents is (generally) the raw file contents, since the metadata is generally much smaller than the file's actual contents. That means that in converting from one format to another, most of the volume's contents doesn't need to change; only the database-ish thing that stores file locations and metadata needs to be converted. So what you can do is build a new-format database-ish thing in the free space in the volume (copying and converting the contents of the old-format one), and then simply say "ok, use the new one now." (and mark the old one as free space in the new format).
That's the oversimplified view; in practice I'm sure there's a lot of complications and fiddly details that need to be dealt with. But I'm pretty sure this is the basic idea.
BTW, there's a similar trick that's sometimes used (including by Apple) to make multiformat (or "hybrid") disks. For example, Apple's CD/DVD burning and image creation software can make disks that're valid as HFS+ volumes (readable on Macs), and also valid ISO 9660 volumes (readable on a wide variety of other OSes that don't know HFS+ from a hole in the ground). It stores one copy of each file's contents, and two different database-ish things (one ISO format, one HFS+), and sets it up so different OSes will see the one they understand and be able to read the volume.