Does it download torrents on your server or web torrent on your browser? - the readme really doesn't say.

Imo downloading on the server is more useful. Web torrent is great but I don't think it's very practical in many places.

Why does this need a server? Isn't the point to be able to add a torrent and start watching immediately?

Browsers can't make torrent connections, or any others for that matter. Except for HTTP and WebRTC.

WebTorrent is a hack to run torrent protocol over WebRTC, but obviously it only connects to other WebTorrent programs and not to normal torrent programs. I think PeerTube uses it.

It could be a desktop app :shrug:

there are several hybrid desktop clients that do support WebTorrent peers