• package it once, instead many times by many different maintainers
• solves the dependency hell
• makes it easier to run multiple versions of same program (or driver) or install a program without it's complete desktop environment
• sandboxed, better control of permissions (at least with Flatpak) and makes easier to backup the whole program version and state
• same package manager across distributions (at least with Flatpak)
• useful on LTS distributions which does not get new packages or programs or even beta software, other than security fixes (think of Debian)
• useful for write only distributions such as SteamOS
• does not need sudo to install new programs (at least with Flatpak and AppImages)

For simple applications this is probably not that wild. But the more complex programs we talk about, the more helpful are these formats. Programs like OBS or Firefox in example is a lot of trouble to compile quickly. And imagine more of these programs. Package maintainer of your distro could use the time in a better way. Those who want to package it themselves (probably Arch) could still do, but most who want to provide the newest Firefox could just use Flatpak, coming directly from the developer day 0.

One also does not need to wait until its packaged by your distro maintainer and it comes directly from the developer instead (maybe). The original developers often do not support all distros and would like to have a known state and version of the program that they can rely on, like a Flatpak.

That being said, I don't use Flatpak. But I used it in the past and it was helpful in some cases. Even on an Arch based distribution. Currently I use an AppImage for a program that is not in the official Arch repos. The AUR has it, but the -bin is outdated and the -git version building from source takes too long and power. Even on my new modern machine it would take at least an hour for every new version. Or I just download the Applmage once (88 MB) and use the self updating system of it (which downloads newest version automatically and renames it to current executable filename). I'm talking about RPCS3 emulator.