They're downloaded somewhere under /var/snap and by default a snap only has access to a limited set of directories - one under /var/snap for system-wide data (generally used by snaps that run services like cups or MySQL) and one under ~/snap for each user. When you snap remove an app, it bundles that up into a file that's kept for a while in case you reinstall, but it won't if you use --purge.

Obviously many apps request access to other places (such as non-hidden directories in your homedir) so they can read or write stuff, but that's down to the app to then behave correctly (same as with any other packaging system).