From the Store description:
Your Windows Phone backs up your text messages, but WP Message Backup is the first app to give you detailed access to those messages. In this beta version of the app, you can browse and export all of the messages you’ve sent or received. It supports normal SMS text messages, picture messages, group messages, video message, and pretty much anything else. You can export messages and conversations as plain text, CSV, or XML.
WP Message Backup does not allow you to send text messages. A Microsoft Account with an associated Windows Phone is required, and that phone must be set to backup text messages (to configure this, look under Backup in Settings).
Here's the app in action - it's not the fastest app in the world - though this is partly to do with Microsoft's servers and their APIs - I had to sit back and wait tens of minutes in order even to be able to do screenshots!
Also, note that the interface is broken in portrait view, IMHO. Far better to run it in landscape view and then you can see all the elements and interact with the UI.
It's a barebones, 'beta'-ish UI throughout, and selection and searching doesn't work properly yet, least of all in portrait mode in my tests...
Things are smoother in landscape mode, mind you, with visibility of messages and the possibility of selecting them...
...though you have to pay a small in-app-purchase if you want to bulk export them. Which is fair enough.
You can grab WP Message Backup here in the Store, it's a free download. It's also early days, so you might want to keep this installed and watch for updates - and also feedback to the developer the issues you may face.