Where can I find the maps my Garmin Edge 305 made?
The Garmin Edge 205/305 Owner's Manual makes it clear that any data that was not saved using the Garmin Training Center Software would be overwritten when the device is full.

The Garmin Training Center Software was last updated Aug 25, 2011. The Windows version ran on XP, but it can still be downloaded by following the links. You'd have to test if it will work on a current OS (but see the next para).

Garmin has a page explaining Where is the history data for Training Center located? That page seems to suggest that the software works on Windows 7.

If the data you want is present on your backup as .TCX files, then you can just upload it to Strava:


  Works for multiple .tcx, .fit or .gpx files 25MB or smaller. Choose up to 25 files.


If you can't see .TCX files then you'll have to dig deeper. From what I see the data can be stored in .TCD files, perhaps compressed a la zip. Install the Garmin Training Center Software and see if it can extract the data you need.

Good luck.