Wow, that is freaky.
If you save the pdf to your PC and then open the pdf with Illustrator, the words and so forth are separate objects and the paper background is also separate.
Here is the BC for anyone who wants to try it:
http://www.whitehouse.gov/sites/default/files/rss_viewer/birth-certificate-long-form.pdf
You can also export images from within Adobe itself. You should get 3 images. The background image (with the green cross hatch) will have the part trimmed away by the clipping mask to be seen. You will see the two dots on the right side that are hidden by the clipping mask.