First off this is different to #4594.
I click the button, nothing happens
After choosing clicking "Open Link", focus should change to Signal and it should start inviting you to the group. (Either adding the group to the menu if its got no joining restrictions, or notify the group admins if it does have restrictions)

v1.39.4
MacOSX Big Sur (v11.1)
Master: Signal-cli (v0.7.0) which does have v2 group support
"Loading..." (Seems to not be working)
Signal Desktop does not currently allow you to join groups via link, but this is something we're working on. Thanks for reporting.
Most helpful comment
Signal Desktop does not currently allow you to join groups via link, but this is something we're working on. Thanks for reporting.