To export your Ruffle save files from a website, just visit the website, then click the bookmarklet in your browser's Bookmarks menu or toolbar. All of the save files should automatically download as .sol files!
The Ruffle desktop program stores its save files at %LocalAppData%\ruffle\SharedObjects. On Windows, you can open this location by pasting it into the Run box (Windows + R).
Here is the easiest way I've found to import save files into the Ruffle desktop program:
Download the SWF file for the game that your save file applies to.
Open the SWF file in Ruffle and play until the game saves your progress (It might happen after you beat the first level, for example).
Open Ruffle's save file folder (as explained previously).
Look through the subfolders until you find a file with the same name as the save file you exported.
Replace the save file with the one you exported. Next time you open the SWF file in Ruffle, your progress should be restored!
Manual Export Tool
Some sites have security settings that prevent bookmarklets from working. The tool below provides a manual method of downloading Ruffle save data as a .sol file. To export your Ruffle save files from a website:
Visit the website and press F12 or Ctrl-Shift-I to open your browser's Developer Tools.
Click the "Application" tab (in Chrome) or the "Storage" tab (in Firefox).
On the left sidebar, expand the "Local Storage" section and select the domain (such as https://example.com)
You will see a list of keys and values. Find the key corresponding to the save file you want to export and copy its corresponding value.
Paste the value into the box below, then enter a name for the save file into the next box.