Most of the backgrounds you see with images and information on the left hand side in the background are generally created in an image program such as Gimp or Photoshop, and then uploaded as a background image. The trick in this method is that you have to have the image scaled to fit correctly in the twitter frame on your computer. You need to remember that everyone uses different size screens with different sized pixels. You unfortunately cannot click on any of these links that are displayed in this fashion. People will need to type them into their browser if they want to go to that page.

I have found some other good resources for creating Twitter Backgrounds.

Check these out too:

The Closet Entrepreneur Lets you know how to use PowerPoint to create a background.

Croncast This will give you specs for creating the size of the image as well as other tips

Presentation Zen This gives a great resource guide to getting quality Stock Photos for your Backgrounds

An example of a Twitter profile creted in Gimp