There are some guides for downloading it onto the local computer (which you mention trying) and uploading to the server.
However, the current requirements include command line access (typically through ssh) and with reason as it permits (with composer)
- Installation
- Updating
- Management of extensions
.. which as I understand it, are otherwise not currently possible without it.
There are plans for not needing ssh access and completing everything through the web UI but they are not completed yet. Look at the github issues for flarum. #880 might be a start
So if you want want to have flarum now in a functional manner, you should consider switching to a hoster that allows command line level access.
If you just want to test it, you can utilize Devflarum