Good news: you can now send URLs in direct messages.
This isn’t a new feature per se, but it’s been disabled by Twitter for quite some time.
For about a year Twitter’s support page on posting direct messages included this paragraph:
We’re restructuring back-end elements of our direct message system. As a result, users may be unable to send some URLs in direct messages. We apologize for the inconvenience.
This has now been removed, as you can see here.
As to why was it disabled in the first place, rumour has it that Twitter was working on a way to filter the massive amounts of spam and malicious links that were sent via DMs. That new system now appears to be in place.
Lately, Twitter has been making noises about revamping the direct message system, and this update might be part of that process.