Are you using a mac, and is it TB Standalone or the Steam version?
If you're using a mac, which is probable (because some users have had problems with it in the past), the steps you have to follow are:
Mac: TB Standalone
1. Download/Save as replay
2. Open 2 windows in Finder
3. Go to Toribash -> Replays sub-folder in Finder
4. Drag replay file from downloads (or whichever place you stored the file in) to the replays sub-folder in your other Finder window
5. Open Toribash and ta-da!
If it's the steam version, it's pretty much the same steps; but instead of going to Finder to have the replays sub-folder open you have to:
Mac: Steam TB version
1. ctrl + r-click the Toribash game
2. Click on properties
3. Head over to local files
4. Click on browse local files
5. Find the replays sub-folder and drag the replay file
If it's not Mac, it should be "open file with: [insert app to open the file]". You would select Toribash and it'd work. If it doesn't work you can follow the steps above more-or-less and it should work also.