It seems they have forgotten to probably set it up, so when you connect through GitHub it tries to redirect to the old website which doesn't work anymore after you authorise to log in through GitHub. When it starts taking long to redirect you back, right click and copy the link where it says "please visit this setup page to continue"
You will get a link like this: https://weblate.itch.ovh/accounts/complete/github/?code=
Just change it from weblate.itch.ovh to weblate.itch.zone, and then put this in your URL bar and then you will be logged in.