AppImages are portable software that don't need root permission to be run. They can be run without root permission:
chmod +x ExamplePackage.AppImage
Most AppImages require sys-fs/fuse:0 on slot zero to be run.
dlopen(): error loading libfuse.so.2
AppImages require FUSE to run. You might still be able to extract the contents of this AppImage if you run it with the --appimage-extract option.
See https://github.com/AppImage/AppImageKit/wiki/FUSE for more information
This error can be solved by installing sys-fs/fuse:0. Slot 0 has version 2 of fuse required by most AppImages, whereas sys-fs/fuse:3 which has version 3 of fuse, this is installed by default if no slot is specified and doesn't solve the dependency problem.
emerge --ask sys-fs/fuse:3