How do I get rid of this rust in the trunk?

If what your pictured is the extent of the rust then it might be savable. I recently did my trunk and it was in similar shape but only a couple places where it was rusted through, mainly in those ribbed areas. I sand blasted mine then treated it with Ospho. After that you can coat it with whatever you desire.
If your spare tire well is in bad shape you might consider replacing the whole trunk floor. Its not that hard to do and may end up being less work than salvaging what you have.
Pic 1 -After sandblasting
Pic 2 -Primered
Pic 3 -2K tinted bedliner, shouldnt rust again for a while.

20170503_215702.jpg

20170527_220324.jpg

20170604_140401.jpg