Browsers block a page opened from disk from calling ChirpStack, because ChirpStack sends no CORS headers. The relay is a bookmark that runs inside a ChirpStack tab, where calls are allowed, and passes them back here.
Drag CS relay to your bookmarks bar once. After that, Connect opens a ChirpStack tab: click the bookmark there. The relay moves into a small window of its own, the ChirpStack tab closes, and you're back here, signed in. Keep the small relay window open (minimizing it is fine). If you're already signed in to ChirpStack in that tab, that session is used here too. After the first connection, clicking the bookmark on a ChirpStack page with no console tab opens the console right there, which is how it works on iPhone and iPad.