By default twitter disables people you don't follow from sending you messages. This has some advantages as well as disadvantages, its purpose is to prevent spam and scam messages but some people need as much access as possible to connect with other people if they are looking for new opportunities, new friends or work etc. So here's how to activate it so anyone can send you messages on twitter.
- Goto Settings -> Privacy and Safety
- Click Direct Messages
- Check on Allow message requests from everyone
Hope this is useful!