Originally, direct messages on Twitter could only be sent between two people who followed each other. Twitter now allows anyone who follows you to send you direct messages, if you enable the feature in your settings.
The new feature started rolling out today. To enable it, head to your account settings page, and scroll down to "Receive direct messages from any follower". It's unchecked by default. To allow your followers to send you DMs without you having to follow them first, tick the box.
The feature could be useful for people who get followers begging them to follow back in order to DM them privately, but it could also lead to a lot of spam — that's why it's off by default.
If that happens, switching the option off is your only recourse. Nonetheless, it's a potentially useful choice.