Good news. You are about to feel very popular. It's so nice when we get a message from someone-well, if it is a nice message that is-and now you can get even more of those on Twitter. Twitter has added a new feature to their Direct Messages which categorizes them either into "Inbox" or "requests." The ones that go into "Requests" are Direct Messages from people you don't follow, but want to connect with you. But you do need to keep your DM open then to preview those and choose whether you want to keep or delete.
If you do accept the message then that user can continue to message you, FYI. Also if someone sends you something NSFW or just plain gross you don't have to look at it or you can just look at it by hitting "view media" but it doesn't mean you fully accept it.
Basically it is a checks and balance system for preventing abuse from users approaching you and whether they are someone you want to connect with or not. However, if it is someone you do want to connect with then this could be a very effective new way to grow your network.