First of all you need Nero in order to actually burn your DVD's. DVD shrink just takes the movie and compresses it.
I believe that you do need to download your movie to your harddrive so dvdshrink can compress the video. Most DVD's are Dual layer discs that can take up as much as 8.5 GB of Hard drive space. You also need up to 4.5 GB of hard drive space so that Dvd Shrink can create your new dvd for your DVD-R. 8.5+4.5=13 GB, so It should barely fit on your hard drive.
Here a quick checklist.
1) Get nero
2) Make sure that you have 13 GB of Hard drive space.