Is your furnace / AC fan running? Is the register made of metal or plastic? It has to be cooled in some manner in order to build up condensation. Maybe there is a cold water pipe touching the duct under the floor. While this is possible it would think this is unlikely. Did you take the register off to check the duct under it? Check it out to see what you find.
When did this start? Has this been happening all along? If it only started a short while ago look for any changes you did around that time. Could be many things but for sure it is colder then the room temperature and what ever moisture there is in the air is condensing on it.
Try to find out what is making it cold.