xoox On the homepage of Flarum, it says 'you don't have to be a sysadmin and says we can simply upload the files on a shared host and be done with it.
The first part is correct already. Using shared hosting is possible, so you don't need to manage your server. At the time being, however, you need SSH access to your shared hosting to run Composer for installation und update processes.
The second goal, having just to upload the files and running them in your browser for installation and update is not yet fulfilled. Such an option is planned once Flarum reaches a non-beta stage, though there are some workarounds available already like Pockethold.
But I have been reading a lot of posts here about how you need a VPS and need Composer and stuff. I am a little confused about this.
Most people associate SSH access with a VPS (and thus having to manage your server instance), but that's not true. Look out for shared host with SSH access, those two terms should give you some search results.
Often, Composer is already installed on theses accounts. If it's not, you can install it easily.