Flatpak is like an alternative packaging system that exists outside of your distro's normal packaging model, e.g. apt/dnf/pacman etc. The killer features are that Flatpaks work on any distro with a single universal package, and that the software versions will be cutting-edge without needing cutting-edge system dependencies. Flatpaks run in their own dependency network and generally don't rely on anything from the host system - this means that you can have arbitrary software on your machine that your distro/repo maintainers don't need to compile/quality-control/stability-test/etc. It also comes with an easy sandboxing framework out of the box as a bonus.

In my case I usually use Flatpaks to get more current versions of software without totally messing up Debian's "Debian does not break" stability model - Debian is meticulously maintained so that its "Stable" branch only has ultra-stable versions of software, at the expense of those packages being older and frozen. If you use a distro with smaller package repos (e.g. OpenSUSE/Fedora/etc) you'll probably appreciate finding Flatpak versions of software that you'd normally need to manually compile.

Flatpaks are cool, and they have a specific use. They're not the end-all be-all of packaging and they're (hopefully) not going to replace apt/dnf/pacman. As for why they hate apt I have no idea. apt is good, and you can even make it a little nicer by installing nala and using that instead of apt.

If the basis of this thread is that you're digging for distro recommendations I'd personally steer you towards Linux Mint and OpenSUSE Tumbleweed for their ease of use. Debian is a little more difficult to set up than Linux Mint but not tremendously so. Arch is more of an "intermediate" difficulty distro where the main challenge is that your system packages are fast-moving and can break/change in small ways from day-to-day. If you aren't comfortable with Linux you might get frustrated with minor bugs that you don't know how to troubleshoot. Conversely, if you want to learn Linux then dealing with Arch's shenanigans will help expose you to various parts of the system naturally.