If you want to transfer your cards from one computer to another, the 'File - Export' functionality is not what you should use, as this is meant for sharing cards with other people and this will not export your own learning data.
In case you want to transfer your data from an old computer you will no longer use, it's enough to backup Mnemosyne's data directory and copy it to your new computer. More info about the location of the directories etc. here.
If you still want to review your cards on both computers, the sync mechanism is the way to go. More info here.