After you login to Bandsintown please click on the artist you want to manage. Then click the Settings button on the bottom left. There you will see options to connect your Twitter account to Bandsintown. Toggle the button On. This will prompt you to grant Bandsintown permission to post on this account for you.
Then when you share a message or event from the Home page of Bandsintown it will also automatically send to your Twitter page if you select Twitter when you send the message.