Either way the dialog above should tell you the full path. You’ll see one of two directories, for Office 2007 versions it’s usually located in the UProof folder, but in older versions it was in the Proof folder.
To backup your dictionary file, browse down to the path you saw above, or just enter in the following into the location bar in explorer: Note that you should use the Edit Word List dialog in here to make any changes (don’t try and make changes to the file itself, I’ve read about people having issues when doing that)
After we transfer the file from the old computer to the new one, you’ll need to use the Add button here to add the dictionary, if it’s not automatically added.
This will bring up the screen where you can assign custom dictionaries, as well as see the full path to the current dictionary file.