By default Aegisub automatically saves a backup copy of every subtitle file you open, along with another copy every minute (if any changes have been made in the last minute). These copies can be found in %APPDATA%\Aegisub\autosave on Windows, ~/.aegisub/autosave on Linux, and ~/Library/Application Support/Aegisub/autosave on OS X. In addition, they can be browsed directly within Aegisub at File → Open Autosaved File…
Aegisub automatically cleans up old files in the autosave directory, so the automatic backups should not be used for long-term storage of files.