Okay.. through some trial and error, I figured it out.
Make sure that you fill in only the domain and not a complete URL. Examples blow:
Thus not: http://www.domain.com or www.domain.com
But just: domain.com
It would be really helpful that the alert gives an error message when you've typed in an URL and doesn't say 'You've added succesfully bla, bla, bla".
Also make sure that you fill in on the settings page a callback URL which is obviously the URL of the website where you use the application. Otherwise the Twitter API can't redirect the user after a succesfull auhorication back to the website, because there is simply no callback URL defined.
I hope this helps anyone.