The solution depends on your operating system. From the Kongregate forums:
Go to Folder Settings in Windows and turn hidden folders ON.
Next, go to
C:\Users\YOUR WINDOWS
USERNAME\AppData\Roaming\Macromedia\FlashPlayer\#SharedObjects\SOME
RANDOM LETTERS AND NUMBERS\
There should be two folders, one named chat.kongregate.com and one named armorgames.com or a different website you played it on
Copy the save files/folders from the source computer to the destination computer
You’re DONE! You can now pick up where you left off, and hopefully make progress towards the new badges (which WILL appear… eventually)
The folder locations for various OS's are as follows:
On Microsoft Windows NT 5.x and 6.x, they are stored in:
%APPDATA%\Macromedia\Flash Player\#SharedObjects\
%APPDATA%\Macromedia\Flash Player\macromedia.com\support\flashplayer\sys\
On Mac OS X, they are stored in:
~/Library/Preferences/Macromedia/Flash Player/#SharedObjects/
~/Library/Preferences/Macromedia/Flash Player/macromedia.com/support/flashplayer/sys/
On Linux or Unix, they are stored in:
~/.macromedia/Flash_Player/#SharedObjects/
~/.macromedia/Flash_Player/macromedia.com/support/flashplayer/sys/
For Linux and Unix systems, if the open-source Gnash plugin is being used instead of the official Adobe Flash, they will instead be found at:
~/.gnash/SharedObjects/
http://en.wikipedia.org/wiki/Local_shared_object#File_locations