"If your DMs are open and someone you don't follow messages you, it will go into requests," the company wrote in an announcement today. "From requests, you can accept or delete that message. If you accept, it will go into your inbox." According to Twitter's updated support page, deleting the message doesn't prevent that user from communicating with you in the future (you'll need to block or mute for that) but it does clear up your message requests. If you've accepted a message from a user in the past, new messages from that user will also go directly into your inbox, even if you still don't follow that person.
All attached media will also be hidden whenever an unknown user sends a direct message, but images and video can still be viewed without accepting the conversation by clicking the "View Media" button. Finally, new group DMs will get the same Accept/Delete prompt if you're being added by a user you don't know, and if you have read receipts turned on for DMs, the sender won't know you've seen their message until you click accept. Of course, thanks to Twitter's API restrictions, the new DM inbox function is only rolling out on Twitter's official iOS and Android apps today. According to Twitter, it should also start showing up on the web version in the next few months.